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^6 



THE 



Imerican house-carpenter : 



A TREATISE UPON 

ARCHITECTURE, • 

CORNICES AND MOULDINGS 

FRAMING, 

DOORS, WINDOWS, AND STAIRS 

TOGETHER WITH 

THE MOST IMVORTAKT PRINCIPLES 

OF 

PRACTICAL GEOMETRY. 



BY R. G. HATFIELD, 

ARCHITECT. 
FIFTH EDITION, WITH ADDITIONS, 



JUustrateD bp more t{)dn V^x^t i)un^reti 2inarc-;biua3 



NEW YORK: 
JOHN WILEY, 16V BROADWAY 

1 8 52. 



X 



.A 



Entered, according to Act of Congress, in tiie year 1844, by 

R. G. HATFIELD, 

in tlift Clerk's Office of the District Court of the Southern District of New York. 



R, CRAIGHEAD, PRINTER, 

53 Vesey Street, X Y 



f\ 



^ 



J 



PREFACE. 



This book is intended for carpenters — for masters, 
journeymen and apprentices. It has long been the 
complaint of this class that architectural books, in- 
tended for their instruction, are of a price so high as 
to be placed beyond their reach. This is owing, in a 
great measure, to the costliness of the plates with 
which they are illustrated : an imnecessary expense, as 
illustrations upon wood, printed on good paper, answer 
every useful purpose. Wood engravings, too, can be 
distributed among the letter- press ; an advantage 
which plates but partially possess, and one of great 
importance to the reader. 

Considerations of this kind induced the author to 
undertake the preparation of this volume. The sub- 
ject matter has been gleaned from works of the first 
authority, and subjected to the most careful examina- 
tion. The explanations have all been written out 
from the figures themselves, and not taken from any 
other work ; and the figures have all been drawn ex- 
pressly for this book. In doing this, the utmost care 
has been taken to make every thing as plain as the 
nature of the case would admit. 



IV PREFACE. 



The attention of the reader is particularly directed to 
the following new inventions, viz : an easy method of 
describing the curves of mouldings through three 
given points ; a rule to determine the projection of 
eave cornices ; a new method of proportioning a cor- 
nice to a larger given one ; a way to determine the 
lengths and bevils of rafters for hip-roofs ; a way to 
proportion the rise to the tread in stairs ; to determine 
the true position of butt-joints in hand-rails ; to find 
the bevils for splayed-work ; a general rule for scrolls, 
&c. Many problems in geometry, also, have been 
simplified, and new ones introduced. Much labour 
has been bestowed upon the section on stairs, in which 
the subject of hand-rjiiling is presented, in many re- 
spects, in a new, and, it is hoped, more practical form 
than in previous treatises on that subject. 

The author has endeavoured to present a fund of 
useful information to the American house-carpenter 
that would enable him to excel in his vocation ; how 
far he has been successful in that object, the book 
itself must determine. 



PREFACE TO THE FIFTH EDITION. 



Since the first edition of this work was pubUshed, I 
have received numerous testimonials of its excellent 
practical value, from the very best sources, viz. from 
the workmen themselves who have used it, and who 
have profited by it. As a convenient manual for 
reference in respect to every question relating either 
to the simpler operations of Carpentry or the more 
intricate and abstruse problems of Geometry, those who 
have tried it assure me that they have been greatly 
assisted in using it. And, indeed, to the true workman, 
there is, in the study of the subjects of which this 
volume treats, a continual source of profitable and 
pleasurable interest. Gentlemen, in numerous instances 
have placed it in the hands of their sons, who have 
manifested a taste for practical studies ; and have also 
procured it for the use of the workmen upon their 
estates, as a guide in their mechanical operations. I 
was not, then, mistaken in my impressions, that a work 
of this kind was wanted ; and this evidence of its 
usefulness rewards me in a measure for the pains 
taken in its preparation. 

R. G. H. 

New York, Oct. 1, 1852. 



TABLE OF CONTENTS. 



INTRODUCTION. 



Articles necessary for drawing, 
To prepare the paper, 



Art. 
•2 
5 



To use the set-square, 
Directions for drawing, 



Art. 
11 
13 



SECT. I.— PRACTICAL GEOMETRY. 



DEFINITIONS. 

Lines, - - . . 17 

Angles, - - - 23 

Angular point, . - - 27 

Polygons, - - - 28 

The circle, - - - 47 

The cone, ... 56 

Conic sections, - - - 58 

The ellipsis, - - - 61 

The cylinder, . - . 68 

PROBLEMS. 

To bisect a line, - - 71 

To erect a perpendicular, - 72 

To let fall a perpendicular, 73 

To erect ditto on end of line, 74 

Six, eight and ten rule, - 74 

To square end of board, - 74 

To square foundations, &c., 74 
To let fall a perpendicular 

near the end of a line, - 75 

To make equal angles, - 76 

To bisect an angle, - . 77 

To .risect a right angle, 78 

To draw parallel lines, - 79 
To divide a line into equal 

parts, ... - 80 

To find the centre of a circle, 81 

To draw tangent to circle, 82 

Do. without using centre, 83 

To find the point of contact, 84 
To draw a circle through three 

given points, - - 85 



To find a fourth point in circle, 86 
To describe a segment of a 
circle by a set-triangle, - 87 
Do. by intersection of lines, 88 
To curve an angle, - 89 

To inscribe a circle within a 

given triangle, - - 90 
To make triangle about circle, 91 
To find the length of a cir- 
cumference, - - 92 
To describe a triangle, hexa- 
gon, &c., ... 93 
To draw an octagon, - 94 
To eight-square a rail, &c., 94 
To describe any polygon in 

a circle, ... 95 

To draw equilateral triangle, 96 
To draw a square by com- 
passes, - - - 97 
To draw any polygon on a 

given line, ... 98 
To form a triangle of required 

size, . . - . 99 

To copy any right-lined figure, 100 
To make a parallelogram 

equal to a triangle, - 101 

To find the area of a triangle, 101 
To make one parallelogram 

equal another, - - 102 
To make one square equal to 

two others, - - - 103 
To find the length of a rafter, 103 



Vlll 



CONTENTS. 



Art. 

To find the length of a brace, 103 
To ascertain the pitch of a 

roof, - - - - 103 
To ascertain the rake of a 

step-ladder, - - - 103 
To describe one circle equal 

to two others, - - 104 

To make one polygon equal 

to two or more, - - 104 
To make a square equal to 

a rectangle, - - 105 

To make a square equal to 

a triangle, - - - 106 
To find a third proportional, 107 
To find a fourth proportional, 108 
To proportion one ellipsis to 

another, - - - 108 
To divide a line as another, 109 
To find a mean proportional, 110 
Definitions of confc sections, 111 
To find the axes of an ellipti- 
cal section, - - - 112 
To find the axes and base of 

the parabola, - - 113 

To find the height, base and 

axes of the hyperbola, - 114 
To find foci of ellipsis, - 115 
To describe an ellipsis with 

a string, - - - 115 

To describe an ellipsis with 

a trammel, - - 116 

To construct a trammel, - 116 
To describe an ellipsis by or- 

dinates, - - - 117 
To trace a curve through 

given points, - - - 117 
To describe an ellipsis by in- 
tersection of lines, - 118 



120 

121 

122 
128 

124 

125 

126 

127 
128 



Art. 

Do. from conjugate diameters, 118 

Do. by intersecting arcs, - 119 

To describe an oval by com- 
passes, - - - 

Do. in the proportion, 7x9, 
5x7, &c., - 

To draw a tangent to an el- 
lipsis, 

To find the point of contact. 

To find a conjugate to the 
given diameter. 

To find the axes from given 
diameters, - - - 

To find axes proportionate to 
given ones, 

To describe a parabola by in- 
tersection of lines, - 

To describe hyperbola by do., 

DEMONSTRATIONS. 

Definitions, axioms, &;c.. ISO. 139 
Addition of angles, - 140 
Equal triangles, - - 141 
Angles at base of isoceles tri- 
angle equal, - - 142 
Parallelograms divided equal- 
ly by diagonal, - - 143 
Equal parallelograms, - 144 
Parallelogram equal triangles, 146 
To make triangle equal poly- 
gon, - - . . 147 
Opposite angles equal, • - 148 
Angles of triangle equal two 

riglit angles, - - - 149 

Corollaries from do., 150. 155 
Ani^le in semi-circle a rijrht 

angle, - . - 156 

Hecatomb problem, - - 157 



SECT. II.— ARCHITECTURE. 



HISTORY. 

Antiquity of its origin, - 159 
Its cultivation among the an- 
cients, - - - 160 
Among the Greeks, - - 161 



Among the Romans, - 162 
Ruin caused by Gollis and 

Vandals, - - - I'^S 

Of the Gothic, . - 164 

Of the Lombard, - - 165 



CONTENTS. 



IX 



Art. 

Of the Byzantine and Oriental, 166 
Moorish, Arabian and Modern 

Gothic, - - - 167 

Of the English, - - 168 
Revival of the art in the sixth 

century, - - - 169 
The art improved in the 14th 

and 15th centuries, - 170 

Roman styles cultivated, 171 

STYLES. 

Origin of different styles, 172 

Stylobate and pedestal, - 173 

Definitions of an order, - 174 
Of the several parts of an 

order, - - 175. 185 

To proportion an order, - 186 

The Grecian orders, - 187 

Origin of the Doric, - - 188 

Intercolumniation, - - 189 

Adaptation, - - - 190 

Origin of the Ionic, - 191 

Characteristics, - - 192 

Intercolumniation, - - 193 

Adaptation, - - - 194 

To describe the volute, - 195 

Origin of the Corinthian, - 196 

Adaptation, - - - 197 

Persians, - - - - 199 

Caryatides, - - - 200 

The Roman orders, - - 202 



Ah. 
Extent of Roman structures, 203 
Roman styles copied from 

Grecian, - - . 208 
Origin of the Tuscan, - 204 

Adaptation, - - . 205 
Characteristics of the Egypt- 
ian, .... 206 
Extent of Egyptian structures, 206 
Adaptation, - - . 207 
Appropriateness of design, 208. 211 
Durable structures, - - 212 
Plans of dwellings, &c., 213 

Directions for designing, 213, 214 

PRINCIPLES. 

Origin of the art, - - 215 
Arrangement and design, - 216 
Ventilation and cleanliness, 217 
Stability, - - - 218 
Ornaments, - - - 219 
Scientific knowledge neces- 
sary, - - - 220 
The foundation, - - 221 
The column, - . - 222 
The wall, - - - 223 
The lintel, - - - 224 
The arch, - - . 225 
The vault, - - - 226 
The dome, - - - 227 
The roof, - - - 228 



SECT. HI.— MOULDINGS, CORNICES, &c. 



MOULDINGS, &C. 

Elementary forms, - - 229 

Characteristics, - - 230 

Grecian and Roman, - - 231 

Profile, - - - 332 

To describe the torus and 

scotia, - - - - 233 

To describe the echinus, 234 

To describe the cavetto, 235 

To describe the cyma-recta, 236 

To describe the cyma-reversa, 237 



Roman mouldings, - 238 

Modern mouldings, - - 239 
Antse caps, - - - 240 

CORNICES. 

Designs, - . - - 241 

To proportion an eave cornice, 242 

Do. from a smaller given 

one, - - - - 243 

Do. from a larger given 

one, . - - . 244 

To find shape of angle-bracket, 245 

To find form of raking cornice, 246 



CONTENTS. 



SECT. IV.— FRAMING. 



Art. 

Laws of pressure, - - 248 

Parallelogram of forces, - 248 
To measure the pressure on 

rafters, - - - 249 
Do. on tie-beams, - 250 
The effect of position, - 251 
The composition of forces, 252 
Best position for a strut, - 253 
Nature of ties and struts, - 254 
To distinguish ties from struts, 255 
Lattice-work framing, - 256 
Direction of pressure in raft- 
ers, - - - - 257 
Oblique thrust of lean-to roofs, 258 
Pressure on floor-beams, - 259 
Kinds of pressure, - - 260 
Resistance t» compression, 261 
Area of post, - - 261 
Resistance to tension, - 262 
Area of suspending piece, 262 
Resistance to cross-strains, 263 
Area of bearing timbers, 263 
Area of stiffest beam, - 264 
Bearers narrow and deep, 265 
Principles of framing, - 266 

FLOORS. 

Single-joisted, - - 267 

To find area of floor-timbers, 268 

Dimensions of trimmers, &c., 269 

Strutting between beams, 270 

Cros8-furring and deafening, 271 

Double floors, - - - 272 

Dimensions of binding-joists, 273 

Do. of bridging.joists, 274 

Do. of ceiling-joists, - 275 

Framed floors, - - - 276 

Dimensions of girders, - 277 

Girders sawn and bolted, - 278 

Trussed girders, - * 279 

Floors in general, - - 280 

PARTITIONS. 

Nature of their construction, 281 

Designs for partitions, - 282 

Superfluous timber, - - 282 

Improved method, - - 283 

Weight of partitioning, - 284 



ROOFS. 

Lateral strains, 
Pressure on roofs, 
Weight of covering, 
•Definitions, 
Relative size of timbers, 



Art 
285 

286 
286 

287 
288 



To find the area of a king- post, 289 
Of a queen-post, - - 290 
Of a tie-beam, - - - 291 
Of a rafter, - - - 292 
Of a straining-beam, - 294 

Of braces, - - - 295 
Of purlins, - - - 296 

Of common rafters, - 297 

To avoid shrinkage, - - 298 
Roof with a built-rib, - 299 
Badly-constructed roofs, - 300 
To find the length and bevils 

in hip-roofs, - - 301 

To find the backing of a hip- 
rafter, ... - 302 

DOMES. 

With horizontal ties, - 303 
Ribbed dome, - - - 304 
Area of the ribs, - - 305 
Curve of equilibrium, - 30^^ 
To describe a cubic parabola, 307 
Small domes for stairways, 308 
To find the curves of the ribs, 309 
To find the shape of the cover- 
ing for spherical domes, 310 
Do. when laid horizontally, 311 
To find an angle-rib, - - 312 

BRIDGES. 

Wooden bridge with tie-beam, 313 

Do. without a tie-beam, 314 

Do. with a built-rib, 315 

Table of least rise in bridges, 315 

Rule for built- ribs, - - 315 

Pressure on arches, - 316 

To form bent-ribs, - - 317 

Elasticity of timber, - 317 

To construct a framed rib, 318 

Width of roadway, &c., - 319 

Stone abutments and piers, 320 

Piers constructed of piles, 321 



CONTENTS. 



XI 



Art. 

Piles in ancient bridges, 321 

Centring for stone bridges, 322 

Pressure of arch-stones, - 322 
Centre without a tie at the 

base, - - - 323 

Construction of centres, - 324 

General directions, - 325 

Lowering centres, - - 326 

Relative size of timbers, - 327 

Short rule for do. - - 328 

Joints between arch-stones, 329 

Do. in elliptical arch, - 330 

Do. in parabolic arch, - 331 



JOINTS. 

Art. 
Scarfing, or splicing, 332. 334 
To proportion the parts, - 335 
Joints in beams and posts, - 336 
Joints in floor-timbers, - 337 
Timber weakened by framing, 338 
Joints for rafters and braces, 339 
Evil of shrinking avoided, - 340 
Proper joint for collar-beam, 341 
Pins, nails, bolts and straps, 342 
Dimensions of straps, - 342 
To prevent the rusting of 
straps, . - . . 342 



SECT, v.— DOORS, WINDOWS, (fee. 



DOORS 

Dimensions of doors, - - 343 
To proportion height to width, 344 
Width of stiles, rails and 

panels, . . . 345 
Example of trimming, - 346 

Elevation of a door and trim- 
mings, - - - 347 
General directions for hang- 
ing doors, - - - 348 



WINDOWS. 

To determine the size, - 349 
To find dimensions of frame, 350 
To proportion box to flap 

shutter, - - - 351 
To proportion and arrange 

windows, - - - 352 

Circular-headed windows, 353 

To find the form of the soffit, 354 

Do. in a circular wall, - 355 



SECT. VI.— STAIRS. 



Their position, - - - 356 

Principles of the pitch-board, 357 
To proportion the rise to the 

tread, - - - 358 

The angle of ascent, - - 359 

Length of steps, - - 360 

To construct a pitch-board, 361 

To lay-out the string, - 362 

Section of step, - - 363 

PLATFORM STAIRS. 

To construct the cylinder, - 364 

To cut the lower edge of do., 365 

To place the balusters, - 366 



To find the moulds for the 

rail, - - - - 367 

Elucidation of this method, 368 

Two other examples, 369, 370 
To apply the mould to the 

plank, - - - 371 

To bore for the balusters, - 372 

Face-mould for moulded rail, 373 

To apply this mould to plank, 374 

To ascertain thickness of stuff, 375 

WINDING STAIRS- 

Flyers and winders, - 376 

To construct winding stairs, 377 



Xll 



CONTENTS. 



Art. 

Timbers to support winding 

stairs, ... - 378 
To find falling-mould of rail, 379 
To find face-mould of do., 380 
Position of butt-joint, - 380 
To ascertain thickness of 

stufl:; - - . - 381 
To apply the mould to plank, 383 
Elucidation of the butt-joint, 384 
Quarter-circle stairs, - 385 
Falling-mould for do., , - 386 
Face-mould for do., - 387 
Elucidation of this method, 388 
To bevil edge of plank, - 389 
To apply moulds without be- 
villing plank, - -. 390 



AH. 

To find bevils for splayed- 

work, - - - 391 

Another method for face- 
moulds, - - - 892 

To apply face-mould to plank, 394 

To apply falling-mould, . 395 

SCROLLS. 

General rule, - - 396 

To describe scroll for rail, 398 

For curtail-step, - - 399 

Balusters under scroll, - 400 

Falling-mould for scroll, - 401 

Face-mould for do., - 402 

Round rails over winders, - 403 

To find form of newel-cap, 404 



SECT. VII.— SHADOWS. 



Inclination of line of shadow, 497 
Shadows on mouldings, 408 

Shadow of a shelf, - - 409 
Of a shelf of varying width, 410 
Of do. with oblique end, 411 

Of an inclined shelf, - 412 

Of do. inclined in section, 413 
Of do. having a curved edge, 414 
Of do. curved in elevation, 415 
Shadow on cylindrical wall, 416 
Do. on inclined wall, - 417 
Shadow of a beam, - - 418 



Shadow in a recess, - 419 

Do. with wall inclined, - 420 

Shadow in a fireplace, . 421 

Shadow of window lintel, - 422 

Shadow of step-nosing, - 423 

Of a pedestal upon steps, - 424 
Of square abacus on column, 425 

Of circular abacus on do. 426 

On the capital of a column, 427 

Of column and entablature, 428 

Shadows on Tuscan cornice, 429 

Reflected light, - - 430 



APPENDIX. 

Page. 

Glossary of Architectural Terms, - - ' - - 3 

Table of Squares, Cubes and Roots, - - • - 14 

Rules for extending the use of the foregoing table, - - 21 

Rule for finding the roots of whole numbers with decimals, - 23 

Rules for the reduction of Decimals, - - - 23 

Table of Areas and Circumferences of Circles, - - - 25 

Rules for extending the use of the foregoing table, - - 28 

Table showing the Capacity of Wells, Cisterns, &c., - -29 

Rules for finding the Areas, &;c., of Polygons, - - 30 

Table of Weights of Materials, - - - - - 31 



INTRODUCTION. 



Art. 1. — A knowledge of the properties and principles of lines 
can best be acquired by practice. Although the various problems 
throughout this work may be understood by inspection, yet they 
will be impressed upon the mind with much greater force, if they 
are actually performed with pencil and paper by the student. 
Science is acquired by study — art by practice : he, therefore, who 
would have any thing more than a theoretical, (which must of 
necessity be a superficial,) knowledge of Carpentry, will attend 
to the following directions, provide himself with the articles here 
specified, and perform all the operations described in the follow- 
ing pages. Many of the problems may appear, at the first read- 
ing, somewhat confused and intricate ; but by making one line 
at a time, according to the explanations, the student will not 
only succeed in copying the figures correctly, but by ordinary 
attention will learn the principles upon which they are based, 
£ind thus be able to make them available in any unexpected case 
to which they may apply. 

2. — The following articles are necessary for drawing, viz : a 
drawing-board, paper, drawing-pins or mouth-glue, a sponge, a 
T-square, a set-square, two straight-edges, or flat rulers, a lead 
pencil, a piece of india-rubber, a cake of india-ink, a set of draw- 
ing-instruments, and a scale of equal parts. 

3. — ^The size of the drawing-board must be regulated accord- 
ing to the size of the drawings which are to be made upon it. 
Yet for ordinary practice, in learning to draw, a board about 16 

I 



4 AMERICAN HOUSE CARPENTER. 

by 20 inches, and one inch thick, will be found large enough, 
and more convenient than a larger one. This board should be 
well-seasoned, perfectly square at the corners, and without 
clamps on the ends. A board is better without clamps, because 
the little service they are supposed to render by preventing the 
board from warping, is overbalanced by the consideration that 
the shrinking of the panel leaves the ends of the clamps project- 
ing beyond the edge of the board, and thus interfering with the 
proper working of the stock of the T-square. When the stufi 
is well-seasoned, the warping of the board will be but trifling ; 
and by exposing the rounding side to the fire, or to the sun, it 
may be brought back to its proper shape. 

4. — For mere line drawings, the paper need not commonly 
be what is called drawing-paper ; as this is rather costly, and 
will, where much is used, make quite an item of expense. 
Cartridge-paper, as it is called, of about 20 by 26 inches, and of 
as good a quality nearly as drawing-paper, can be bought for 
about 50 cts. a quire, or 2 pence a sheet ; and each sjieet may be 
cut in halves, or even quarters, for practising. If the drawing 
is to be much used, as working drawings generally are, cartridge- 
paper is much better than the other kind. 

5. — A drawing-pin is a small brass button, having a steel pin 
projecting from the under side. By having one of these at each 
corner, the paper can be fixed to the board ; but this can be done 
in a much better manner with mouth-glue. The pins will pre- 
vent the paper from changing its position on the board ; but, 
more than this, the glue keeps the paper perfectly tight and 
smooth, thus making it so much the more pleasant to work on. 

To attach the paper with mouth-glue, lay it with the bottom 
side up, on the board ; and with a straight-edge and penknife, 
cut oflf the rough and uneven edge. With a sponge moderately 
wet, rub all the surface of the paper, except a strip around the 
edge about half an inch wide. As soon as the glistening of the 
water disappears, turn the sheet over, and place it upon the 



INTRODUCTION. 3 

board just where you wish it glued. Commence upon one of 
the longest sides, and proceed thus : lay a flat ruler upon the 
paper, parallel to the edge, and within a quarter of an inch of it 
With a knife, or any thing similar, turn up the edge of the papei 
against the edge of the ruler, and put one end of the cake o1 
mouth-glue between your lips to dampen it. Then holding it 
upright, rub it against and along the entire edge of the paper 
that is turned up against the ruler, bearing moderately against 
the edge of the ruler, which must be held firmly with the left 
hand. Moisten the glue as often as it becomes dry, until a 
sufficiency of it is rubbed on the edge of the paper. Take 
away the ruler, restore the turned-up edge to the level of the 
board, and lay upon it a strip of pretty stiff paper. By rubbing 
upon this, not very hard but pretty rapidly, with the thumb nail 
of the right hand, so as to cause a gentle friction, and heat to be 
imparted to the glue that is on the edge of the paper, you will 
make it adhere to the board. The other edges in succession 
must be treated in the same manner. 

Some short distances along one or more of the edges, may 
afterwards be found loose : if so, the glue must again be applied, 
and the paper rubbed until it adheres. The board must then be 
laid away in a warm or dry place ; and in a short time, the sur- 
face of the paper will be drawn out, perfectly tight and smooth, 
and ready for use. The paper dries best when the board is laid 
level. When the drawing is finished, lay a straight-edge upon 
the paper, and cut it from the board, leaving the glued strip still 
attached. This may afterwards be taken off by wetting it freely 
with the sponge ; which will soak the glue, and loosen the 
paper. Do this as soon as the drawing is taken off, in order that 
the board may be dry when it is wanted for use again. Care 
must be taken that, in applying the glue, the edge of the paper 
does not become damper than the rest : if it should, the paper 
must be laid aside to dry, (to use at another time,) and another 
sheet be used in its place. 



4 AMERICAN HOUSE CARPENTER. 

Sometimes, especially when the drawing board is new, the 
paper will not stick very readily ; but by persevering, this diffi- 
culty may be overcome. In the place of the mouth-glue, a 
strong solution of gum-arabic may be used, and on some 
accounts is to be preferred ; for the edges of the paper need not 
be kept dry, and it adheres more readily. Dissolve the gum in 
a sufficiency of warm water to make it of the consistency of 
linseed oil. It must be applied to the paper with a brush, when 
the edge is turned up against the ruler, as was described for the 
mouth-glue. If two drawing-boards are used, one may be in use 
while the other is laid away to dry ; and as they may be cheaply 
made, it is advisable to have two. The drawing-board having 
a frame around it, commonly called a panel-board, may afford 
rather more facility in attaching the paper when this is of the 
size to suit ; yet it has objections which overbalance that con 
sideration. 

6. — A T-square of mahogany, at once simple in its construc- 
tion, and affording all necessary service, may be thus made. 
Let the stock or handle be seven inches long, two and a quarter 
inches wide, and three-eighths of an inch thick: the blade, 
twenty inches long, (exclusive of the stock,) two inches wide, 
and one-eighth of an inch thick. In joining the blade to the 
stock, a very firm and simple joint may be made by dovetailing 
it — as shown at Fig. 1. 




Fig. L 



INTRODUCTION. 



7. — The set-square is in the form of a right-angled triangle ; 
and is commonly made of mahogany, one-eighth of an inch in 
thickness. The size that is most convenient for general use, is 
six inches and three inches respectively for the sides which con • 
tain the right angle ; although a particular length for the sides is 
by no means necessary. Care should be taken to have the square 
corner exactly true. This, as also the T-square and rulers, 
should have a hole bored through them, by which to hang them 
upon a nail when not in use. 

8. — One of the rulers may be about twenty inches long, and 
the other six inches. The pencil ought to be hard enough to 
retain a fine point, and yet not so hard as to leave ineffaceable 
marks. It should be used lightly, so that the extra marks that 
are not needed when the drawing is inked, may be easily rubbed 
off with the rubber. The best kind of india-ink is that which 
will easily rub off upon the plate ; and, w^hen the cake is rub- 
bed against the teeth, will be free from grit. 

9. — The drawing-instruments may be purchased of mathe- 
matical instrument makers at various prices : from one to one 
hundred dollars a set. In choosing a set, remember that the 
lowest price articles are not always the cheapest. A set, com- 
prising a sufficient number of instruments for ordinary use, well 
made and fitted in a mahogany box, may be purchased at Pike 
and Son's, (Broadway, near Maiden-lane, N. Y.,) for three or four 
dollars. The compasses in this set have a needle point, which 
is much preferable to a common point. 

10.— The best scale of equal parts for carpenters' use, is one 
that has one-eighth, three-sixteenths, one-fourth, three-eighths, 
one-half, five-eighths, three-fourths, and seven-eighths of an 
inch, and one inch, severally divided into twelfths, instead ot 
being divided, as they usually are, into tenths. By this, if it be 
required to proportion a drawing so that every foot of the object 
represented will upon the paper measure one-fourth of an mch, 
use that part of the scale which is divided into one-fourths of an 



" AMERICAN HOUSE-CARPENTER. 

inch, taking for every foot one of those divisions, and for every 
inch one of the subdivisions into twelfths ; and proceed in like 
manner in proportioning a drawing to any of the other divisions 
of the scale. An instrument in the form of a semi-circle, called a 
protractor, and used for laying down and measuring angles, is 
of much service to surveyors, but not much to carpenters. 

ll.-In drawing p^allel lines, when they are to be parallel 
to either side of the board, use the T-square ; but when it is 
required to draw lines parallel to a line which is drawn in a 
direction oblique to either side of the board, the set-square must 
be used. Let a 6, {Fig. 2,) be a line, parallel to which it is 




Fiff. 2. 



desired to draw one or more lines. Place any edge, as c d, of 
the set-square even with said line ; then place the ruler, g h, 
against one of the other sides, as c e, and hold it firmly;' slide 
the set-square along the edge of the ruler as far as it is desired, 
as at/ ; and a line drawn by the edge, if, will be parallel to a b. 
12.— To draw a line, as k I, {Fig. 3,) perpendicular to another, 
as a b, set the shortest edge of the set-square at the line, a b; 
place the ruler against the longest side, (the hypothenuse of the 
right-angled triangle ;) hold the ruler firmly, and slide the set- 
square along until the side, e d touches the point, k; then the 
line, I k, drawn by it, will be perpendicular to a b. In like 



INTRODUCTION. 



manner, the drawing of other problems may be facilitated, as will 
be d' tcovered in using the instruments. 




Fig. 3. 



13. — In drawing a problem, proceed, with the pencil sharpened 
» a point, to lay down the several lines until the whole figure is 
completed; observing to let the lines cross each other at the 
several angles, instead of merely meeting. By this, the length 
of every line will be clearly defined. With a drop or two of 
water, rub one end of the cake of ink upon a plate or saucer, 
imtil a sujSiciency adheres to it. Be careful to dry the cake of 
ink ; because if it is left wet, it will crack and crumble in pieces. 
With an inferior camePs-hair pencil, add a little water to the 
ink that was rubbed on the plate, and mix it well. It should be 
diluted sufiiciently to flow freely from the pen, and yet be thick 
enough to make a black line. With the hair pencil, place a 
little of the ink between the nibs of the drawing-pen, and screw 
the nibs together until the pen makes a fine line. Beginning 
with the curved lines, proceed to ink all the lines of the figure ; 
being careful now to make every line of its requisite length. If 
they are a trifle too short or too long, the drawing will have a 
ragged appearance ; and this is opposed to that neatness and 
accuracy which is indispensable to a good drawing. When the 
ink is dry, efiace the pencil-marks with the india-rubber. If 



8 AMERICAN HOUSE-CARPENTER. 

the pencil is used lightly, they will all rub off, leaving those 
lines only that were inked. 

14. — In problems, all auxiliary lines are drawn light ; while 
the lines given and those sought, in order to be distinguished at 
a glance, are made much heavier. The heavy lines are made 
so, by passing over them a second time, having the nibs of the 
pen separated far enough to make the lines as heavy as desired. 
If the heavy lines are made before the drawing is cleaned with 
the rubber, they will not appear so black and neat ; because the 
india-rubber takes away part of the ink. If the drawing is a 
ground-plan or elevation of a house, the shade-lines, as they are 
termed, should not be put in until the drawing is shaded; as 
there is danger of the heavy lines spreading, when the brush, in 
shading or coloring, passes over them. If the lines are inked 
with common writing-ink, they will, however fine they may be 
made, be subject to the same evil ; for which reason, india-ink 
is the only kind to be used. 



THE 

AMERICAN HOUSE-CARPENTER. 



SECTION I.— PRACTICAL GEOMETRY. 



DEFINITIONS. 



15. — Geometry treats of the properties of magnitudes. 

16. — A -point has neither length, breadth, nor thickness. 

17. —A line has length only. 

18. — Superficies has length and breadth only. 

19. — A plane is a surface, perfectly straight and even in every 
direction ; as the face of a panel when not warped nor winding. 

20. — A solid has length, breadth and thickness. 

21. — A right ^ or straight^ line is the shortest that can be 
drawn between two points. 

22. — Parallel lines are equi-distant throughout their length. 

23. — An angle is the inclination of two lines towards one 
another. i^Fig. 4.) 




Fig. 4. Fig. 5. Fig. & 

2 



10 



AMERICAN HOUSE-CARPENTER. 



24. — A right angle has one line perpendicular to the other. 

{Fig. 5.) 

25. — An oblique angle is either greater or less than a right 

angle. [Fig- 4 and 6.) 

26. — An acute angle is less than a right angle. [Fig. 4.) 

27. — An obtuse angle is greater than a right angle. [Fig. 6.) 

When an angle is denoted by three letters, the middle one, in 
the order they stand, denotes the angular point, and the other 
two the sides containing the angle ; thus, let a 6 c, [Fig. 4,) be 
the angle, then b will be the angular point, and a b and b c will 
be the two sides containing that angle. 

28. — A triangle is a superficies having three sides and angles. 

[Fig. 7, 8, 9 and 10.) 





Fig. 7. 



Fig. & 



29. — An equi-lateral triangle has its three sides equal. 
{Pig. 7.) 

30. — An isoceles triangle has only two sides equal. [Fig. 8.) 
31. — A scalene triangle has all its sides unequal. [Fig. 9) 





Fix. 9. 



Fig. 10. 



32. — A right-angled triangle has one right angle. [Fig. 10.) 

33. — An acute-angled triangle has all its cingles acute. 
[Fig. 7 and 8.) 

34. — An obtuse-angled triangle has one obtuse angle. 
[Fig. 9.) 

35. — \ quadrangle has four sides and four angles. [Fig, 11 
to 16.) 



PRACTICAL GEOMETRY. 



11 



Fig. 11. 



Fig. 12. 



36. — A parallelogram is a quadrangle having its opposite 
sides parallel. {Fig. 11 to 14.) 

37. — A rectangle is a parallelogram, its angles being right 
angles. {Fig. 11 and 12.) 

38. — A square is a rectangle having equal sides. {Fig. 11.) 

39. — A rhombus is an equi-lateral parallelogram having ob- 
lique angles. {Fig. 13.) 




Fig. 13. 



Fig. 14. 



40. — A rhomboid is a parallelogram having oblique angles. 
(Fig. 14.) 

41. — A trapezoid is a quadrangle having only two of its sides 
parallel. {Fig. 15.) 




Fig. 15. 



Fig. 16. 



42. — A trapezium is a quadrangle which has no two of its 
sides parallel. {Fig. 16.) 

43. — A polygon is a figure bounded by right lines. 

44. — A regular polygon has its sides and angles equal. 

45. — An irregular polygon has its sides and angles unequal, 

46. — A trigon is a polygon of three sides, {Fig. 7 to 10 ;) 
a tetragon has four sides, {Fig. 11 to 16 ;) a pentagon has 



12 



AMERICAN HOUSE-CARPENTER. 



five, {Fig, 17 ;) a hexagon six, {Fig. 18 ;) a heptagon seven, 
{Fig. 19 ;) an octagon eight, {Fig. 20 ;) a nonagon nine ; a 
decagon ten ; an undecagon eleven ; and a dodecagon twelve 
sides. 




Fig. 17. 



Fig. 18. 





Fig. 19. 



Fig. 20. 



47. — A circle is a figure bounded by a curved line, called the 
circumference ; which is every where equi-distant from a cer- 
tain point within, called its centre. 

The circumference is also called the periphery/, and sometimes 
the circle. 

48. — The radius of a circle is a right line drawn from the 

centre to any point in the circumference, {a 6, Fig. 21.) 

All the radii of a circle are equal. 




Fi-. 21. 



49. — The diameter is a right line passing through the centre, 
and terminating at two opposite points in the circumference. 
Hence it is twice the length of the radius, (c d, Fig. 21.) 

50. — An arc of a circle is a part of the circumference, (c b or 
bed, Fig. 21.) 

51. — A chord is a right line joining the extremities of an arc 
(6 d, Fig. 21.) 



PRACTICAL GEOMETRY. 



13 



52. — ^A segment is any part of a circle bounded by an arc and 
its chord. {A, Fig. 21.) 

53. — A sector is any part of a circle bounded by an arc and 
two radii, drawn to its extremities, (i?, Fig, 21.) 

54. — A quadrant^ or quarter of a circle, is a sector having a 
quarter of the circumference for its arc. (C, Fig. 21.) 

55. — A tangent is a right line, which in passing a curve, 
touches, without cutting it. [f g^ Fig. 21.) 

56. — A cone is a solid figure standing upon a circular base 
diminishing in straight lines to a point at the top, called its 
vertex. {Fig. 22.) 




Fig. 22. 




57.— The axis of a cone is a right line passmg through it, from 
the vertex to the centre of the circle at the base. 

58. — An ellipsis is described if a cone be cut by a plane, not 
parallel to its base, passing quite through the curved surface, 
(a 6, Fig. 23.) 

59. — A parabola is described if a cone be cut by a plane, 
parallel to a plane touching the curved surface, (c rf, Fig. 23 — 
c d being parallel to / g.) 

60. — An hyperbola is described if a cone be cut by a plane, 
parallel to any plane within the cone that passes through its 
vertex, (e A, Fig. 23.) 

61. — Foci are the points at which the pins are placed in de- 
scribing an ellipse. (See Art. 115, and/,/, Fig, 24.) 



14 



AMERICAN HOUSE-CARPENTER. 




62. — The transverse axis is the longest diameter of the 
ellipsis, [a b, Fig. 24.) 

63. — The conjugate axis is the shortest diameter of the 
ellipsis ; and is, therefore, at right angles to the transverse axis. 
(c d, Fig. 24.) 

64. — The parameter is a right line passing through the focus 
of an ellipsis, at right angles to the transverse axis, and termina- 
ted by the curve, {g h and g t, Fig. 24.) 

65. — A diameter of an ellipsis is any right line passing 
through the centre, and terminated by the curve, {k I, or m n, 
Fig. 24.) 

66. — A diameter is conjugate to another when it is parallel to 
a tangent drawn at the extremity of that other — thus, the diame- 
ter, m n, {Fig. 24,) being parallel to the tangent, o p, is therefore 
conjugate to the diameter, k I. 

67. — A double ordinate is any right line, crossing a diameter 
of an ellipsis, and drawn parallel to a tangent at the extremity of 
that diameter, {i t, Fig. 24.) 

68. — A cylinder is a solid generated by the revolution of a 
right-angled parallelogram, or rectangle, about one of its sides ; 
and consequently the ends of ^Lie :ylinder are equal circles. 
(Fig. 25.) 



PRACTICAL GEOMETRY. 



15 




Fig. 25. 



Fig. 26. 



69. — The axis of a cylinder is a right line passing through it, 
from the centres of the two circles which form the ends. 

70. — A segment of a cylinder is comprehended under three 
planes, and the curved surface of the cylinder. Two of these 
are segments of circles : the other plane is a parallelogram, called 
by way of distinction, the plane of the segment. The circular 
segments are called, the ends of the cylinder. {Fig. 26.) 



PROBLEMS. 



RIGHT LINES AND ANGLES. 



Tl. — To bisect a line. Upon the ends of the hne, a 6, {Fig. 
^5) as centres, with any distance for radius greater than hall 




a b, describe arcs cutting each other in c and d ; draw the line, 

c d, and the point, e, where it cuts a b, will be the middle of the 

line, a b. 

In practice, a line is generally divided with the compasses, or 
dividers ; but this problem is useful where it is desired to draw, 
at the middle of another line, one at right angles to it. (See 
Art. 85.) 

d 




a 
Fig. 28. 



72, — To erect a perpendicular* From the point, a, {Fig* 28,) 



PRACTICAL GEOMETRY. 



ir 



set oflf any distance, as a 6, and the same distance from a to c ; 
upon c, as a centre, with any distance for radius greater than c a, 
describe an arc at d ; upon 6, with the same radius, describe 
another at d ; join d and a, and the Une, d a, will be the per- 
pendicular required. 

This, and the three following problems, are more easily per- 
formed by the use of the set-square — (see Art. 12.) Yet they 
are useful when the operation is so large that a set-square cannot 
be used. 




Fig. 29. 



73. — To let fall a 'perpendicular. Let a, [Fig. 29,) be the 
point, above the line, h c, from which the perpendicular is re- 
quired to fall. Upon a, with any radius greater than a d, de- 
scribe an arc, cutting 6 c at e and/; upon the points, e and/, 
with any radius greater than e d, describe arcs, cutting each 
other at g ; join a and g, and the line, a d, will be the perpen- 
dicular required. 




Fig. 30. 



74. — To erect a perpendicular at the end of a line. Let a, 
(Fig. 30,) at the end of the line, c a, be the point at which the 
perpendicular is to be erected. Take any point, as 6, above the 

3 



18 



AMERICAN HOUSE-CARPENTER. 



line, c a, and with the radius, b a, describe the arc, d a e, 

through d and Z>, draw the hue, d e ; join e and a, then e a will 

be the perpendicular required. 

The principle here made use of, is a very important one ; and 
is applied in many other cases — (see Art. 81, 6, and Art. 84. 
For proof of its correctness, see Art. 156.) 




74, a. — A second method. Let &, {Fig. 31,) at the end of th* 
line, a b, be the point at which it is required to erect a perpen- 
dicular. Upon 6, with any radius less than b a, describe the arc, 
c e d ; upon c, with the same radius, describe the small arc at e, 
and upon e, another at d ; upon e and d, with the same or any 
other radius greater than half e d, describe arcs intersecting at / ; 
join/ and 6, and the line,/ 6, will be the perpendicular required. 




d 

Fig. 32. 



74^ 5. — A third method. Let b, {Fig. 32,) be the given point 
at which it is required to erect a perpendicular. Upon b, with any 
radius less than b a, describe the quadrant, d ef; upon d, with 
the same radius, describe an arc at e, and upon e, another at c ; 



PRACTICAL GEOMETRY. 19 

through d and e, draw d c, cutting the arc in c ; join c and 6, 
then c h will be the perpendicular required. 

This problem can be solved by the six^ eight and ten rule, 
as it is called ; which is founded upon the same principle as 
the problems at Art. 103, 104 ; and is applied as follows. Let 
a c?, [Fig. 30,) equal eight, and a e, six ; then, \i d e equals ten, 
the angle, e a c?, is a right angle. Because the square of six 
and that of eight, added together, equal the square of ten. thus : 
6 X 6 = 36, ai*d 8 X 8 = 64 ; 36 + 64 = 100, and 10 x 10 - 
100. Any sizes, taken in the same proportion, as six, eight and 
ten, will produce the same effect : as 3, 4 and .5, or 12, 16 and 
20. (See note to Art. 103.) 

By the process shown at Fig. 30, the end of a board may be 
squared without a carpenters'-square. All that is necessary is a 
pair of compasses and a ruler. Let c a be the edge of the board, 
and a the point at which it is required to be squared. Take the 
point, 6, as near as possible at an angle of forty-five degrees, or on 
ami^re-line, from a, and at about the middle of the board. This 
is not necessary to the working of the problem, nor does it affect 
its accuracy, but the result is more easily obtained. Stretch the 
compasses from h to a, and then bring the leg at a around to d ; 
draw a line from c?, through 6, out indefinitely ; take the dis- 
tance, d b, and place it from b to e ; join e and a ; then e a will 
be at right angles to c a. In squaring the foundation of a build- 
ing, or laying-out a garden, a rod and chalk-line may be used 
instead of compasses and ruler. 

75. — To let fall a perpendicular near the end of a line. 

Let e, {Fig. 30,) be the point above the line, c a, from which the 

perpendicular is required to fall. From e, draw any line, as e c?, 

obliquely to the line, c a ; bisect e d at b ; upon 6, with the 

radius, b e, describe the arc, e a d ; join e and a ; then e a will 

be the perpendicular required. 




76. — To make an angle, (as e df Fig. 33,) eqnal to a given 
angle, (as b a c.) From the angular point, a, with any radius 
describe the arc, be; and with the same radius, on the line, d e, 



20 



AMERICAN HOUSE-CARPENTER. 



and from the point, c?, describe the arc,/^; take the distance, 

b c, and upon g, describe the small arc at/; join /and d ; and 

the angle, e df, will be equal to the angle, b a c. 

If the given line upon which the angle is to be made, is situa- 
ted parallel to the similar line of the given angle, this may be 
performed more readily with the set-square. (See Art. 11.) 




Fig. 34. 



77. — To bisect an angle. Let a b c, {Fig. 34,) be the angle 

to be bisected. Upon b, with any radius, describe the arc, a c ; 

upon a and c, with a radius greater than half a c, describe arcs 

cutting each other at d ; join b and d ; and b d will bisect the 

angle, a 6 c, as was required. 

This problem is frequently made use of in solving other pro- 
blems ; it should therefore be well impressed upon the memory. 




Fig. 35. 

78. — To trisect a right angle. Upon a, {Fig. 35,) with any 

tadius, describe the arc, b c ; upon b and c, with the same radius, 

describe arcs cutting the arc, 6 c, at c? and e ; from d and e, draw 

lines to a, and they will trisect the angle as was required. 

The truth of this is made evident by the following operation. 
Divide a circle into quadrants : also, take the radius in the divi- 
ders, and space off the circumference. This will divide the 
circumference into just six parts. A semi-circumference, there- 



PRACTICAL GEOMETRY. ^tl 

fore, is equal to three, and a quadrant to one and a half of those 
parts. The radius, therefore, is equal to f of a quadrant ; and 
this is equal to a right angle. 






Fig. 3G. 

79. — Through a given pointy to draw a line parallel to a 
given line. Let a, {Fig. 36,) be the given point, and b c the 
given line. Upon any point, as </, in the line, h c, with the 
radius, d a, describe the arc, a c; upon a, with the same radius, 
describe the arc, d e ; make d e equal to a c ; through e and o, 
draw the line, e a ; which will be the line required. 

This is upon the same principle as Art. 76. 




80. — To divide a given line into any number of equal parts. 

Let a bj {Fig. 37,) be the given line, and 5 the number of parts. 

Draw a c, at any angle to a b ; on a c, from a, set off 5 equal 

parts of any length, as at 1, 2, 3, 4 and c ; join c and b ; through 

the points, 1, 2, 3 and 4, draw 1 e, 2/, 3 ^ and 4 h, parallel to 

c b ; which will divide the line, a -6, as was required. 

The lines, a b and a c, are divided in the same proportion. 
(See Art. 109.) 

THE CIRCLE. 

81. — To find the centre of a circle. Draw any chord, as a b, 



22 



4MERICAN HOUSE-CARPENTER. 




{Fig. 38,) and bisect it Avith the perpendicular, c d ; bisect c d 
with the line, e f, as at g ; then g is the centre as was required. 




81, a. — A second method. Upon any two points in the cir- 
cumference nearly opposite, as a and h. {F\s:. 39,) describe arcs 
cutting each other at c and d : take any other two points, as e 
and/, and describe arcs intersecting as at g and h ; join g and A, 
and c and d ; the intersection, o, is the centre. 

This is upon the eame princi|)!e as Art. 85. 




81, b. — A third method. Draw any chord, as a b, (Pig, 40,) 



PRACTICAL GEOMETRY. 23 

and from the point, a, draw a c, at right angles to a b ; join 

c and b ; bisect c 6 at a — which will be the centre of the circle. 

If a circle be not too large for the purpose, its centre may very 
readily be ascertained by the help of a carpenters'-square, thus : 
app' y the corner of the square to any point in the circumference, 
as at a ; by the edges of the square, (which the lines, a b and 
a c, represent,) draw lines cutting ihe circle, as at b and c ; join 
b and c ; then if 6 c is bisected, as at c?, the point, d, will be the 
centre. (See Art. 156.) 




82. — At a given jjoiiil in a circle, to draw a tangent thereto. 
Let a, {Fig, 41,) be the given point, and b the centre of the cir- 
cle. Join a and b ; through the point, a, and at right angles to 
a bj draw c d ; c dis the tangent required. 




Fig. 42. 



S3. — The same, without making use of the centre of the 
circle. Let a, {Fig. 42,) be the given point. From a, set oiF 
any distance to 6, and the same from b to c ; join a and c ; 
upon a, with a b for radius, describe the arc, d b e ; make d b 
equal to be; through a and d, draw a line ; this will be the 
tangent required. 

84. — A circle and a tangent given, to find the jpoint of con- 
tact. From any point, as a, {Fig, 43,) in the tangent, b c, draw 



u 



AAIERJCA X 1 :(irSl>CARPB:NTER. 




a line to the centre d ; bisect a d at e ; ^ipon e, with the radius, 

e a, describe the arc, afd;fis the point of contact required. 

If/ and d were joined, the line would form right angles with 
the tangent, b c. (See A/^t. 156.) 

b 




Fig. 44. 



85. — Through any three points not in a straight line, to 
draw a circle. Let a, h and c, [Fig. 44,) be the three given 
points. Upon a and b, with any radius greater than half a b, 
describe arcs intersecting at d and e ; upon b and c, with any 
radius greater than half b c, describe arcs intersecting at /and g ; 
through d and e, draw a right line, also another through /and g ; 
upon the intersection, A, with the radius, h a, describe the cn*cie, 
ah c, and it will be the one required. 




Fig. 45. 



PRACTICAL GEOMETRY. 



26 



86. — Three points not in a straight line being given, to find 

a fourth that shall, with the three, lie in the circumference of 

a circle. Let a b c, {Fig. 45,) be the given points. Connect 

them with right lines, forming the triangle, a c b ; bisect the 

angle, cb a, {Art. 77,) with the line, b d ; also bisect c ain e, 

and erect e d, perpendicular to a c, cutting b d in d ; then d is 

i\iQ fourth point required. 

A fifth point may be found, as at/, by assuming a, d and b, 
as the three given points, and proceeding as before. So, also, 
any number of points may be found ; simply by using any three 
already found. This problem will be serviceable in obtaining 
short pieces of very flat sweeps. (See Art. 311.) 




87. — To describe a segment of a circle by a sei-triaiigle. 
Let a b, {Fig. 46,) be the chord, and c d the height of the seg- 
ment. Secure two straight-edges, or rulers, in the position, c e 
and cf by nailing them together at c, and afiixing a brace from 
e to/; put in pins at a and b ; move the angular point, c, in 
the direction, a c b ; keeping the edges of the triangle hard 
against the pins, a and b ; a pencil held at c will describe the 
arc, a c b. 

If the angle formed by the rulers at c be a right angle, the 
segment described will be a semi-circle. This problem is useful 
in describing centres for brick arches, when they are required to 
be rather flat. Also, for the head hanging- stile of a window- 
frame, where a brick arch, instead of a stone lintel, is to be 
placed over it. 



g 1 



1 h s 



:^^rr 1^^:^; 



3 (2 3 

Fig. 47. 

4 



26 



AMERICAN HOUSE-CARPENTER. 



88. — To describe the segment of a circle hy intersection of 

lines. Let a b, {Fig. AT,) be the chord, and c d the height of 

the segment. Through c, draw ef parallel to a b ; draw 6 /at 

right angles to c b ; make c e equal to c /; draw a g and b h, 

at right angles to a b ; divide c e, c f d a, d b, a g and b h, each 

into a like number of equal parts, as four ; draw the lines, 1 1, 

2 2, &c., and from the points, o, o and o, draw lines to c ; at the 

intersection of these lines, trace the curve, a cb, which will be 

the segment required. 

In very large work, or in laying out ornamented gardens, &c., 
this will be found useful ; and where the centre of the proposed 
arc of a circle is inaccessible, it will be invaluable. (To trace 
the curve, see note at Art. 117.) 




Fig. 48. 



89. — In a given angle, to describe a tanged curve. Let a 

b c, {Fig. 48,) be the given angle, and 1 in the line, a b, and 5 

in the line, b c, the termination of the curve. Divide 1 b and b 5 

into a like number of equal parts, as at 1, 2, 3, 4 and 5 ; join 1 

and 1, 2 and 2, 3 and 3, &c. ; and a regular curve will be formed 

that will be tangical to the line, a b, at the point, 1, and to 6 c 

at 5. 

This is of much use in stair-building, in easing the angles 
formed between the wall-string and base of the hall, also between 
the front string and level facia, and in many other instances. 
The curve is not circular, but of the form of the parabola, {Fig. 
93 ;) yet in large angles the difference is not perceptible. This 
problem can be applied to describing segments of circles for door- 




PRACTICAL GEOMETRY. 



2T 



heads, window-heads, &c., to rather better advantage than Art, 
87. For instance, let a b, {Fig. 49,) be the width of the open- 
ing, and c d the height of the arc. Extend c d^ and make d e 
equal to t, d ; join a and e, also e and b ; and proceed as direct- 
ed at Arf 89. 




Vv:. 



90. — fo describe a circle within any given triangle^ so that 
the sides of the triangle shall be tangical. Let a b c, [i'^ig- 
50,) be the given tiiangle. Bisect the angles, a and 6, according 
to Art. 77 ; upon d, the point of intersection of the bisecting 
lines, with the radius d e, describe ihe required circle. 




91. — About a given circle, to describe an equi-lateral tri- 
angle. Let a d b c, {Fig. 51 ,) be the given circle. Draw the 
diameter, c d ; upon d, with the radius of the given circle, de- 
scribe the arc, a e b ; join a and b ; drsiW f g, at right angles to 
d c ; make/c and c g, each equal to a b ; from/, through a, 
draw / h, also from gj through 6, draw g h; thenfg h will be 
the triangle required. 



28 



AMERICAN HOUSE-CARPENTER. 




g P 



92. — To find a right line nearly equal to the circumference 
of a circle. Let ah c d^ [Fig. 52,) be the given circle. Draw 
the diameter, a c ; on this erect a.n equi-lateral triangle, a e c, 
according to Art. 96 ; draw gf parallel to a c ; extend e c to/, 
also e ato g ; then ^/ will be nearly the length of the semi- 
circle, a d c ; and twice g f will nearly equal the circumference 
of the circle, ab c d^as was required. 

Lines drawn Irom e, through any points in the circle, as o, o 
and 0, top, p and/>, will divide gf in the same way as the semi- 
circle, a d c, is div^ioed. So, any portion of a circle may be 
transferred to a straight line Tni> i : \ very useful problem, 
and should be well studied ; as it is frequently used to solve 
problems on stairs, domes, <5lc. 




Fig. 53. 



92, a. — Another method. Let a bf c, {Pig. 53,) be the given 
circle. Draw the diameter, a c ; from d, the centre, and at right 
angles to a c, draw d b ; join b and c ; bisect be at e; from rf, 
through e, draw df; then e/ added to three times the diameter, 



PRACTICAL GEOMETRY. 



29 



will equal the circumference of the circle within the 4^1)0 part of 
its length. 

POLYGONS, &c. 

93. — Within a given circle^ to inscribe an equi-lateral tri- 
angUy hexagon or dodecagon. Let abed, {Fig. 54,) be the 




Ti., 



given circle. Draw the diaaieter, b d ; upon 6, with the radius 

v»f the given circle, dcFcribe the arc, a e c ; join a and c, also a 

eind d, and c and d — and the triangle is completed. For the 

I;e*xagon : from a, also from c, through e, draw the lines, a f 

and eg; join a and b, b and c, c and/, (fcc, and the hexagon is 

completed. The dodecagon may be formed by bisecting the 

sides of the hexagon. 

Each side of a regular hexagon is exactly equal to the radius 
of the circle that circumscribes the figure. For the radius is 
equal to a chord of an arc of 60 degrees ; and, as every circle is 
supposed to be divided into 360 degrees, there is just 6 times 60, 
vxr 6 arcs of 60 degrees, in the whole circumference. A line 
c'rawn from each angle of the hexagon to the centre, (as in the 
figure,) divides it into six equal, equi-lateral triangles. 




30 



AMERICAN HOUSE-CARPENTER. 



94. — Within a square to inscribe an octagon. Let a b c d, 
{Fig. 55,) be the given square. Draw the diagonals, a d and 
6c; upon a, 6, c and c?, with a e for radius, describe arcs cut- 
ting the sides of the square at 1, 2, 3, 4, 5, 6, 7 and 8 ; join 1 
and 2, 3 and 4, 5 and 6, (fcc, and the figure is completed. 

In order to eight-square a hand-rail, or any piece that is to be 
afterwards rounded, draw the diagonals, a d and b c, upon the 
end of it, after it has been squared-up. Set a gauge to the dis- 
tance, a e, and run it upon the whole length of the stuff, from 
each corner both ways. This will show how much is to be 
chamfered oif, in order to make the piece octagonal. 




Fif?. 56. 



Fig. 57. 




95. — Within a given circle to inscribe any regular polygon. 

Let a b c 2, [Fig. 56, 57 and 58,) be given circley. Draw the 

diameter, a c ; upon this, erect an equi-lateral triangle, a e c, 

according to Art. 96 ; divide a c into as many equ.^'. parts as the 

polygon is to have sides, as at 1, 2, 3, 4, (fee. ; ficii e, through 

each even number, as 2, 4, 6, &c., draw lines cuttmg the circle 

in the points, 2, 4, (fee. ; from these points and at right angles to 

a c, draw lines to the opposite part of the circle ; this will give 

the remaining points for the polygon, as b, /, (fee. 

In forming a hexagon, the sides of the triangle erected upon 
a c, (as at Fig. 57,) mark the points, b and/. 

96. — Upon a given line to construct an equi-lateral triangle. 

Let a bj {Fig. 59,) be the given line. Upon a and b, with a b 



PRACTICAL GEOMETRY. 



31 




for radius, describe arcs intersecting at c ; join a and c, also c 
and b ; then a cb will be the triangle required. 




Fig. 60 

97. — To describe an equi-latei^al j^ectangle, or square. Let 
a b, {Fig. 60,) be the length of a side of the proposed square. 
Upon a and b, with a b for radius, describe the arcs, a d and be; 
bisect the arc, a e, in / ; upon e, with e f for radius, describe the 
arc, c f d ; join a and c, c and d^ d and b ; then a c db will 
be the square required. 




Fig- 61. 



Fig. 62. 



Fig. 63. 



98. — Upon a given line to describe any regular polygon. 
Let a 6, {Fig. 61, 62 and 63,) be given lines, equal to a side of 
the required figure. From 6, draw b c, at right angles to a 6 ; 
upon a and b, with a b for radius, describe the arcs, a c d and 



32 AMERICAN HOUSE-CARPENrER. 

f eh ] divide a c into as many equal parts as the polygon is to 

have sides, and extend those divisions from c towards d ; from 

the second point of division counting from c towards a, as 3, 

{Fig, 61,) 4, {Fig. 62,) and 5, {Fig. 63,) draw a line to b ; take 

the distance from said point of division to a, and set it from b 

to e ; join e and a ; upon the intersection, o, with the radius, 

a. describe the circle, a f d b ; then radiating lines, drawn 

from b through the even numbers on the arc, a c?, will cut the 

circle at the several angles of the required figure. 

[n the hexagon, {Fig. 62,) the divisions on the arc, a d, are 
not necessary ; for the point, o, is at the intersection of the arcs, 
a d and/ 6, the points, /and d, are determined by the intersec- 
tion of those arcs with the circle, and the points above, g and A, 
can be found by drawing lines from a and b, through the centre, 
0. In polygons of a greater number of sides than the hexagon, 
the intersection, o, comes above the arcs ; in such case, therefore, 
the lines, a e and h 5, {Fig. 63,) have to be extended before they 
will intersect. 




Fiff. 64. 



99. — To construct a triangle whose sides shall be severally 
equal to three given lines. Let a, h and c, {Fig. 64,) be the 
given lines. Draw the line, d e, and make it equal to c ; upon 
e, wit}i b for radius, describe an arc at/; upon c?, with a for 
radius, describe an arc intersecting the other at/; join d and/, 
also/ and e ; then dfe will be the triangle required. 




Fig. 65. Fig. 66. 



PRACTICAL GEOMETRY. 



S3 



100. — To construct a figure equal to a given^ right-lined 

figure. Let abed, {Fig. 65,) be the given figure. Make ef, 

{Fig. 66,) equal to c d ; upon /, with d a for radius, describe an 

arc at g ; upon e, with c a for radius, describe an arc intersecting 

the other at g ; join g and e ; upon / and g, with d b and a b 

for radius, describe arcs intersecting at h ; join g and A, also h 

and/; then Fig. 66 will every way equal Fig. 65. 

So, right-lined figures of any number of sides may be copied, 
oy first dividing them into triangles, and then proceeding as 
above. The shape of the floor of any room, or of any piece of 
land, (fee, may be accurately laid out by this problem, at a scale 
upon paper ; and the contents in square feet be ascertained by 
the next. 




101. — To make a parallelogram equal to a given triangle. 

Let ab c, {Fig. 67,) be the given triangle. From a, draw a rf, 

at right angles to be; bisect a d in e ; through e, draw fg^ 

parallel to b c ; from b and c, draw b f and c g, parallel to d e ; 

then bfgc will be a parallelogram containing a surface exactly 

equal to that of the triangle, a b c. 

Unless the parallelogram is required to be a rectangle, the lines, 
b f and c g, need not be drawn parallel to d e. If a rhomboid is 
desired, they may be drawn at an oblique angle, provided they 
be parallel to one another. To ascertain the area of a triangle, 
multiply the base, b c, by half the perpendicular height, da. In 
doing this, it matters not which side is taken for base. 



A ^/ 

^^ e 
^^ C 



i 
Fig. 68. 

5 



34 



AMERICAN HOUSE-CARPENTER. 



102. — A parallelogram being given, to construct another 
equal to it, and having a side equal to a given line. Let A^ 
{Fig. 68,) be the given parallelogram, and B the given line 
Produce the sides of the parallelogram, as at a, 6, c and d ; make 
e d equal to B ; through c?, draw c /, parallel to g b ; through 
e, draw the diagonal, c a ; from a, draw a /, parallel to e d 
then C will be equal to A. (See Art. 144.) 



A 


a ^"V,^, 




B 



Fig 69. 

103. — To make a square equal to two or fjtore given squares. 
Let A and B, {Fig. 69,) be two given squares. Place them so 
as to form a right angle, as at a ; join b and c ; then the square, 
C, formed upon the line, b c, will be equal in extent to the squares, 
A and B, added together. Again : if a b, {Fig. 70,) be equal to 




the side of a given square, c a, placed at right angles to a b, be the 
side of another given square, and c d, placed at right angles to 



PRACTICAL GEOMETRY. 35 

c 6, be the side of a third given square ; then the square, A, 
formed upon the line, d 6, will be equal to the three given 
squares. (See Art. 157.) 

The usefulness and importance of this problem are proverbial. 
To ascertain the length of braces and of rafters in framing, the 
length of stair-strings, &c., are some of the purposes to which it 
may be applied in carpentry. (See note to Art. 74, b.) If the 
length of any two sides of a right-angled triangle is known, that 
of the third can be ascertained. Because the square of the 
hypothenuse is equal to the united squares of the two sides that 
contain the right angle. 

(1.) — The two sides containing the right angle being known, 
to find the hypothenuse. Rule. — Square each given side, add 
the squares together, and from the product extract the square- 
root : this will be the answer. For instance, suppose it were 
required to find the length of a rafter for a house, 34 feet wide,— 
the ridge of the roof to be 9 feet high, above the level of the 
wall-plates. Then 17 feet, half of the span, is one, and 9 feet, 
the height, is the other of the sides that contain the right angle. 
Proceed as directed by the rule : 



17 
17 




9 
9 


119 
17 


- 


81 = square of 9. 
289 = square of 17. 


289 - 


= square of 17. 


370 Product. 



1 ) 370 ( 19-235 + = square-root of 370 ; equal 19 feet, 2; in. 
1 1 nearly : which would be the required 

— length of the rafter. 

29 ) 270 
9 261 



382)- -900 
2 764 



3843 ) 13600 
3 11529 



38465)- 207100 
192325 



(By reference to the table of square-roots in the appendix, the 
root ot almost any number may be found ready calculated.) 



36 AMERICAN HOUSE-CARPENTER. 

Ag^ir : suppose it be required, in a frame building, to find the 
length of a brace, having a run of three feet each way from the 
poixit oi" the right angle. The length of the sides containing the 
right angle will be each 3 feet : then, as before — 

3 
3 

9 = square of one side. 
3 times 3=9 = square of the other side. 

] 8 Product : the square-root of which is 4*2426 -f ft., 
or 4 feet, 2 inches and :ths. full. 

(2.) — The hypothenuse and one side being known, to find the 
other £ide. Rnle. — Subtract the square of the given side from 
the square of tlie hypothenuse, and the square-root of the product 
will be the answer. Suppose it were required to ascertain the 
greatest perpendicular height a roof of a given span may have, 
when pieces of timber of a given length are to be used as rafters. 
Let the span be 20 feet, and the rafters of 3x4 hemlock joist. 
These come about 13 feet long. The known hypothenuse, 
then, is 13 feet, and the known side, 10 feet — that being half the 
span of the building. 

13 
13 



39 
13 



169 = square of hypothenuse. 
10 times 10 = 100 = square of the given side. 



69 Product : the square-root of which is 8 
•3066 + feet, or 8 feet, 3 inches and ^ths. full. This will be 
the greatest perpendicular height, as required. Again : suppose 
that in a story of 8 feet, from floor to floor, a step-ladder is re- 
quijed, the strings of which are to be of plank, 12 feet lone ; and 
it is desirable to know the greatest run such a length of string 
will aflford. In this case, the two given sides are — hypothenuse 
12, perpendicular 8 feet. 

12 times 12 ^ 144 = square of hypothenuse. 
8 times 8 = 64 = square of perpendicular. 

80 Product : the square-root of which is 8*9442 -f 
feet, or 8 feet, 11 inches and i^gths. — the answer, as re<iuire4« 



PRACTICAL GEOMETRY. 



3f 



Many other cases might be adduced to show the utility of this 
problem. A practical and ready method of ascertaining the 
length of braces, rafters, (fcc, when not of a great length, is to 
apply a rule across the carpenters'-square. Suppose, for the 
length of a rafter, the base be 12 feet and the height 7. Apply 
the rule diagonally on the square, so that it touches 12 inches 
from the corner on one side, and 7 inches from the corner on the 
other. The number of inches on the rule, which are intercepted 
by the sides of the square, 13^ nearly, will be the length of the 
rafter in feet ; viz, 13 feet and |ths of a foot. If the dimensions 
are large, as 30 feet and 20, take the half of each on the sides of 
the square, viz, 15 and 10 inches ; then the length in inches 
across, will be one-half the number of feet the rafter is long. 
This method is just as accurate as the preceding ; but when 
the length of a very long rafter is sought, it requires great care 
and precision to ascertain the fractions. For the least variation 
on the square, or in the length taken on the rule, would make 
perhaps several inches difference in the length of the rafter. 
For shorter dimensions, however, the result will be true enough. 




104. — To make a circle equal to tioo given circles. Let .4 
and i?, [Fig. 71,) be the given circles. In the right-angled tri- 
angle, ah c, make a b equal to the diameter of the circle, B, and 
c b equal to the diameter of the circle, A ; then the hypothenuse. 




rig. 72. 



3a 



AMERICAN HOUSE-CARPENTER. 



a Cy will be the diameter of a circle, C, which will be equal in 

area to the two circles, A and B, added together. 

Any polygonal figure, as A, {Fig. 72,) formed on the hypo- 
thenuse of a right-angled triangle, will be equal to two similar 
figures,* as B and C, formed on the two legs of the triangle. 



^ ~^ 


T h 


f '/^ 


\ 


xV 


J 



Fig. 73 

105. — To construct a square equal to a given rectangle. 
Let A, [Fig. 73,) be the given rectangle. Extend the side, a 6, 
and make b c equal to 6 e; bisect a c in/, and upon/, with the 
radius, / «, describe the semi-circle, age; extend e b, till it 
cuts the curve in g ; then a square, b g h d, formed on the line, 
b gy will be equal in area to the rectangle, A. 



B 



(Z 


\ 


A 


b 



Fig. 74. 



g 



105, a. — Another method. Let A, {Fig. 74,) be the given 
rectangle. Extend the side, a 6, and make a d equal to a c ; 

• Similar figures are such as have their several angles respectively equal, and their 
sides respectively proportionate. 



PRACTICAL GEOMETRY. 39 

bisect a d in e ; upon e, with the radius, e a, describe the semi- 
circle, a f d ; extend^ h till it cuts the curve in/; join a and 
/; then the square, B^ formed on the line, a/, will be equal in 
area to the rectangle. A, (See Art. 156 and 157.) 

106. — To form a square equal to a given triangle. Let a 6, 
{Fig. 73,) equal the base of the given triangle, and h e equal 
half its perpendicular height, (see Fig. 67 ;) then proceed as 
directed at Art. 105. 




107. — Two right lines being given, to find a third propor- 
tional thereto. Let A and B, {Fig. 75,) be the given lines. 
Make a h equal to A ; from a, draw a c, at any angle with ab ; 
make a c and a d each equal to B ; join c and b ; from d, draw 
d e, parallel to c b ; then a e will be the third proportional re- 
quired. That is, a e bears the same proportion to B, as B does 
to A. 




108. — Three right lines being given, to find a fourth pro 
portional thereto. Let A, B and C, {Fig. 7^,) be the given 
lines. Make a b equal to A ; from a, draw a c, at any angle 
with a b; make a c equal to B, and a e equal to C ; join c and 
b ; from e, draw e f parallel to c 6 ; then a f will be the fourth 
proportional required. That is, a f bears the same proportion 
to C, as B does to A. 



40 



AMERICAN HOUSE-CARPENTER. 



To apply this problem, suppose the two axes of a given ellipsis, 
and the longer axis of a proposed ellipsis are given. Then, by 
this problem, the length of the shorter axis to the proposed ellip- 
sis, can be foimd ; so that it will bear the same proportion to the 
longer axis, as the shorter of the given ellipsis does to its longer. 
(See also, Art. 126.) 




109. — A li7ie with certain divisions being given, to divide 

another, longer or shorter, given line in the same proportion. 

Let A, {Fig. 77,) be the line to be divided, and B the line with 

its divisions. Make a b equal to B, with all its divisions, as at 

1,2, 3, &c. ; from a, draw a c, at any angle with a b ; make a c 

equal to A ; join c and b ; from the points, 1, 2, 3, &c., draw 

lines, parallel Xo c b ; then these will divide the line, a c, in the 

same proportion as B is divided — as was required. 

This problem will be found useful in proportioning the mem- 
bers of a proposed cornice, in the same proportion as those of a 
given cornice of another size. (See Art. 243 and 244.) So of 
a pilaster, architrave, (fee. 




110. — Between two given right lines, to find a mean pro- 
portional. Let A and /?, [Fig. 78,) be the given lines. On 
the line, a c, make a b eqwul to A, and b c equal to B ; bisect a 
c me ; upon e, with e a for radius, describe the semi-circle, a d 



PRACTICAL GEOMETRY. 



41 



c ; at 6, erect b d, at right angles to a c ; then b d will be the 
mean proportional between A and B. 

For an application of this problem, see Art. 105. 

CONIC SECTIONS. 

111. — If a cone, standing upon a base that is at right angles 
with its axis, be cut by a plane, perpendicular to its base and 
passing through its axis, the section will be an isoceles triangle ; 
(as a b c^ Fig. 79 ;) and the base will be a semi-circle. If a 




cone be cut by a plane in the direction, e/, the section will be 
an ellipsis ; if in the direction, m I, the section will be a para- 
bola ; and if in the direction, r o, an hyperbola. (See Arl. 56 
to 60.) If the cutting planes be at right angles with the plane, 
a 6 c, then — 

112.— To find the axes of the ellipsis, bisect e /, [Fig. 79,) 
in g ; through g, draw h i, parallel to a b ; bisect hiinj ; upon 
j, with j h for radius, describe the semi-circle, h k i ; from g, 
draw g kj at right angles to hi ; then twice g k will be the 
conjugate axis, and e f the transverse. 

6 



42 



AMERICAN HOUSE-CARPENTER. 



113. — To find the axis and base of the parabola. Let m /, 
{Fig. 79,) parallel to a c, be the direction of the cutting plane. 
From m, draw m d, at right angles to a b ; then I m will be the 
axis and height, and m 6^ an ordinate and half the base ; as at 
Fig. 92, 93. 

114. — To find the height ^ base and transverse axis of an 
hyperbola. Let o r, {Fig. 79,) be the direction of the cutting 
plane. Extend o r and a c till they meet at n ; from o, draw 
p, at right angles to a b; then r o Avill be the height, nr the 
transverse axis, and o p half the base ; as at Fig. 94. 




Fig. 80. 



115. — The axes being given, to find the foci, and to describe 

an ellipsis ivith a string. Let a b, {Fig. 80,) and c d, be the 

given axes. Upon c, with a e ovb e for radius, describe the arc, 

ff; then /and/, the points at which the arc cuts the transverse 

axis, will be the foci. At/ and /place two pins, and another at c ; 

tie a string about the three pins, so as to form the triangle, //c ; 

remove the pin from c, and place a pencil in its stead ; keeping the 

string taut, move the pencil in the direction, eg a; it will then 

describe the required ellipsis. The lines, fg and g f show the 

position of the string when the pencil arrives at g. 

This method, when performed correctly, is perfectly accurate; 
but the string is liable to stretch, and is, therefore, not so good to 
use as the trammel. In making an ellipse by a string or twine, 
that kind should be used which has the least tendency to elasticity. 
For this reason, a cotton cord, such as chalk-lines are commonly 
made of, is not proper for the purpose : a linen, or flaxen cord is 
much better. 



PRACTICAL GEOMKTRV, 43 




116. — The axes being given, to describe an ellipsis mith a 
trammel. Let a b and c d, {Fig. 81,) be the given axes. Place 
the trammel so that a line passing tlirongh the centre of the 
grooves, would coincide with the axes ; make the distance from 
the pencil, e, to the nut,/, equal to half c d ; also, from the pen- 
cil, e, to the nut, g^ equal to half a b ; letting the pins under the 
nuts slide in the grooves, move the trammel, e g, in the direction, 
c b d ; then the pencil at e will describe the required ellipse. 

A trammel maybe constructed thus : take two straight strips of 
board, and make a groove on their face, in the centre of their 
width ; join them together, in the middle of their length, at right 
angles to one another ; as is seen at Fig. 81. A rod is then to be 
prepared, having two moveable nuts made of v/ood, with a mor- 
tice through them of the size of the rod, and pins under them 
large enough to fill the grooves. Make a hole at one end of the 
rod, in which to place a pencil, in the absence of a regular tram- 
mel, a temporary one may be made, Vv'hich, for any short job, 
will answer every purpose. Fasten two straight-edges at right 
angles to one another. Lay them so as to coincide with the axes 
of the proposed ellipse, having the angular point at the centre. 
Then, in a rod having a hole for the pencil at one end, place two 
brad-awls at the distances described at Art. 116. While the 
pencil is moved in the direction of the curve, keep the brad-awls 
hard against the straight-edges, as directed for using the tram- 
mel-rod, and one-quarter of the ellipse will be drawn. Then, 
by shifting the straight-edges, the other three quarters in succes- 
sion may be drawn. If the required ellipse be not too large, a 
carpenters'-square may be made use of, in place of the straight- 
edges. 

An improved method of constructing the trammel, is as fol- 
lows : make the sides of the grooves bevilling from the face of 
the stuff, or dove-tailing instead of square. Prepare two slips of 
wood, each about two inches long, which shall be of a shape to 
just fill the groove when slipped in at the end. These, instead of 



44 



AMERICAN HOUSE-CARPENTER. 



pins, are to be attached one to each of the moveable nuts with 
a screw, loose enough for the nut to move freely about the screw 
as an axis. The advantage of this contrivance is, in preventing 
the nuts from slipping out of their places, during the operation 
of describing the curve. 




Fiff.82. 



117. — To describe an ellipsis by ordinates. Let a b and c d, 

[Fig. 82,) be given axes. With c e on e d for radius, de- 
scribe the quadrant,/^ h ; divide f h^ a e and e 6, each into a 
like number of equal parts, as at ] , 2 and 3 ; through these 
points, draw ordinates, parallel io c d andfg- ; take the distance, 
1 i, and place it at 1 /, transfer 2 ; to 2 m, and 3 k to3 n; through 
the points, a, n^ 7n, I and c, trace a curve, and the ellipsis will 
be completed. 

The greater the number of divisions on a e, &c., in this and 
the following problem, the more points in the curve can be found, 
and the more accurate the curve can be traced. If pins are 
placed in the points, n, 7n, Z, (fcc, and a thin slip of wood bent 
around by them, the curve can be made quite correct. This 
method is mostly used in tracing face-moulds for stair hand- 
railini 



'ir>' 




118. — To describe an ellipsis by intersection of lines. Let 



PRACTICAL GEOMETRY. 



45 



a b and c d, {Fig. 83,) be given axes. Through c, draw f g, 
parallel to a b ; from a and b, draw a / and b g, at right angles 
to ab ; divide f a, g b, a e and e 6, each into a like number of 
equal parts, as at 1, 2, 3 and o, o, o ; from 1, 2 and 3, draw lines 
tocy through o, o and o, draw lines from c?, intersecting those 
drawn to c ; then a curve, traced through the points, i, i, i, will 
be that of an ellipsis. 




Where neither trammel nor string is at hand, this, perhaps, is 
the most ready method of drawing an ellipsis. The divisions 
should be small, where accuracy is desirable. By this method, 
an ellipsis may be traced without the axes, provided that a diame- 
ter and its conjugate be given. Thus, a b and c d, {Fig- 84,) are 
conjugate diameters : f g is drawn parallel to a 6, instead of 
being at right angles to c d ; also, / a and g b are drawn paraUel 
to c dj instead of being at right angles to a b. 




119. — To describe an ellipsis by intersecting arcs. Let a h 



46 



AMERICAN HOUSE-CARPENTER. 



and c d, {Fig. 85,) be given axes. Between one of the foci, / 
and/, and the centre, e, mark any number of points, at random, 
as 1, 2 and 3 ; upon /and/, with b 1 for radius, describe arcs at 
^) ^5 §" ^^^ S ; upon/ and/ with a 1 for radius, describe arcs inter- 
secting the others at^,^,^ and g ; then these points of intersection 
will be in the curve of the ellipsis. The other points, h and i, are 
found in like manner, viz : h is found by taking h 2 for one radius, 
and a 2 for the other ; i is found by taking b 3 for one radius, and 
a 3 for the other, always using the foci for centres. Then by 
tracing a curve through the points, c, g, h, i, 6, &c., the ellipse 
will be completed. 

This problem is founded upon the same principle as that of the 
string. This is obvious, when we reflect that the length of the 
string is equal to the transverse axis, added to the distance between 
the foci. See Fig. 80; in Avhich c/ equals a e, the half of the 
transverse axis. 



/I 


'\ ^\ 




^^ 




i 

^ 


^ 


i/ "/ 





g 

Fiff. 8G. 



120. — To describe a figure nearly in the shape of an ellip- 
sis^ by a pair of compasses. Let a b and c d, {Fig. 86,) be 
given axes. From c, draw c e, parallel to ab ; from a, draw a e, 
parallel to c d; join e and c?; bisect e a in/; join / and c, inter- 
secting e dini; bisect i cin o ; from o, draw og, at right angles 
to i c, meeting c d extended to g ; join i and g, cutting the trans- 
verse axis in r ; make h j equal to h g, and h k equal to h r ; 
from J, through r and A:, draw j m and j n; also, from g, through 
k, araw g I; upon g and j, with g c for radius, describe the 



PRACTICAL GEOMETRY. 



47 



arcs, i I and m n ; upon r and k, with r a for radius, describe 
the arcs, m i and I n ; this will complete the figure. 

When the axes are proportioned to one another as 2 to 3, the 
extremities, c and d, of the shortest axis, will be the centres for 
describing the arcs, ^ Zand m ?i; and the intersection of e d with 
the transverse axis, will be the centre for describing the arc, m i, 
&c. As the elliptic curve is continually changing its course from 
that of a circle, a true ellipsis cannot be described with a pair of 
compasses. The above, therefore, is only an ^ipproximation. 




121. — To draw an oval in the jjroportion^ seven by nine. 
Let c c?, {Fig. 87,) be the given conjugate axis. Bisect c d in o, 
and through o, draw a b, at right angles to c d ; bisect c o in e ; 
upon 0, with o e for radius, describe the circle, e f g- h ; from e, 
through h and/, draw ej and e i; also, from g, through h and/, 
draw g k and g I ; upon g, with g c for radius, describe the arc, 
kl ; upon e, with e d for radius, describe the arc, j i ; upon h and 
/ with h k for radius, describe the arcs, j k and I i; this will 
complete the figure. 

This is a very near approximation to an ellipsis ; and perhaps no 
method can be found, by which a well-shaped oval can be drawn 
with greater facility. By a little variation in the process, ovals 
of different proportions may be obtained. If quarter of the trans- 
verse axis is taken for the radius of the circle, efg A, one will be 
drawn in the proportion, five by seven. 



46 



AMERICAN HOUSE-CARPENTER. 




122. — To draw a tangent to an ellipsis. Let abed, {Fig. 
88j) be the given ellipsis, and d the point of contact. Find the 
foci, [Art. 115,)/ and/, and from them, through d, draw/e and 
f d ; bisect the angle, [Art. 77,) e d o, with the line, 5 r ; then 
s r will be the tangent required. 




c Fig. 89. 



123. — An ellipsis with a tangent given, to detect the point 
of contact, hetagbf, {Fig. 89,) be the given ellipsis and tan- 
gent. Through the centre, e, draw a h, parallel to the tangent j 
any where between e and/, draw c d, parallel to ab ; bisect c d in 
; through o and e, dmwfg; then g will be the point of con- 
tact required. 

124. — A diameter of an ellipsis given, to find its conjugate. 
Let a b, {Fig. 89,) be the given diameter. Find the line,/^, by 
the last problem; then/^ will be the diameter required. 



PRACTICAL GEOMETRY. 



49 




125. — An]/ diameter and its conjugate being given, to as- 
certain the two axes, a7id thence to describe the ellipsis. Let 
a b and c d, {Fig. 90,) be the given diameters, conjugate to one 
another. Through c, draw e /, parallel to a b ; from c, draw c 
g, at right angles to ef; make c g equal to a h or h b ; join g 
and h ; upon g, with g c for radius, describe the arc, i k c j ; 
upon A, with the same radius, describe the arc, I n ; through the 
int*"rsections, I and n, draw n o, cutting the tangent, ef, in o ; 
upon 0, with o gfov radius, describe the semi-circle, eig f ; join 
e and^, also g and/, cutting the arc, i c j, in k and t ; from e, 
through h, draw e 7?i, also from/, through A, draw//? ; from A; 
and ^, draw k r and ^ 5, parallel to^ A, cutting e ni in r, and/p 
in s ; make A ni equal to A r, and h p equal to h s ; then r m 
and 5 /> will be the axes required, by which the ellipsis may be 
drawn in the usual way. 

126. — To describe an ellipsis, whose axes shall be propor- 
tionate to the axes of a larger or smaller given one. Let a 
cbd, {Fig. 91,) be the given ellipsis and axes, and i j the trans- 
verse axis of a proposed smaller one. Join a and c ; trom i, 
Iraw i e, parallel to a c ; make o f equal to oe ; then e/ will be 



5C 



AMERICAN HOUSE-CARPENTER. 




the conjugate axis required, and will bear the same proportion to 
ij, asc d does to a b. (See Art. 108.) 




2 3 m 3 2 
Fi-. 92. 



127. — To describe a parabola by intersection of lines. Lee 
m Ij {Fig. 92,) be the axis and height, (see Fig. 79,) and d dy & 
double ordinate and base of the proposed parabola. Through / 
draw a a, parallel to d d ; through d and d, draw d a and d a, 
parallel to ni I ; divide a d and d m^ each into a like number of 
equal parts ; from each point of division in d m, draw the lines, 
1 1, 2 2, (fee, parallel Xoml; from each point of division in d 
a, draw lines to I ; then a curve traced through the points of 
intersection, o, o and o, will be that of a parabola. 

127, a. — Another method. Let rn, I, {Fig. 93,) be the axis and 
height, and d d the base. Extend m I, and make I a equal to m 
I /join a and d, and a and d ; divide a d and a d. each into a 
like number of equal parts, as at 1, 2, 3, &c. ; join 1 and 1, 2 and 
2, &c., and the parabola will be completed. 



PRACTICAL GEOMETRY. 



51 



iff 


T 


3/ 


V 


a/ 


V 


1/ 


V 




m 
Fig. 93. 



j>l2.<o3 'Z L p 

Fig. 94. 



128. — To describe an hyperbola by intersection of lines. 
Let r 0, (Fig. 94,) be the height, p p the base, and n r the trans- 
verse axis. (See Fig. 79.) Through r, draw a a, parallel to p 
p ; from p, draw a p, parallel to r ; divide a p and p 0, each 
into a like number of equal parts ; from each of the points of di- 
visions in the base, draw lines to n ; from each of the points of 
division in a p, draw lines to r ; then a curve traced through the 
points of intersection, 0, 0, (fcc, will be that of an hyperbola. 

The parabola and hyperbola afford handsome curves for various 
mouldings. 



^DEMONSTRATIONS. 



129. — To impress more deeply upon the mind of the learner 
some of the more important of the preceding problems, and to 
indulge a veiy common and praiseworthy curiosity to discover 
the cause of things, are some of the reasons why the following 
exercises are introduced. In all reasoning, definitions are ne- 
cessary ; in order to insure, in the minds of the proponent and 
respondent, identity of ideas. A corollary is an inference deduced 
from a previous course of reasoning. 'An ac^.iom is a proposition 
evident at first sight. In the following demonstrations, there are 
many axioms taken for granted ; (such as, things equal to the 
same thing are equal to one another, &c. ;) these it was thought 
not necessary to introduce in form. 



b 
Fie. 95. 



130. — Definition. If a straight line, as a 6, [Fig. 95,) stt.nd 
upon another straight line, as c d^ so that the two angles made at 



PRACTICAL GEOMETRY. 53 

the point, 6, are equal — a b do a b d, (see note to Art. 27,) then 
each of the two angles is called a right angle. 

131. — Defifiition. The circumference of every circle is sup- 
posed to be divided into 360 equal parts, called degrees ; hence 
a semi-circle contains 180 degrees, a quadrant 90, <fec. 




132. — Definition. The measure of an angle is the number of 
degrees contained between its two sides, using the angular point 
as a centre upon which to describe the arc. Thus the arc, c e, 
{Fig. 96,) is the measure of the angle, c b e ; e a, of the angle, 
e b a ; and a d, of the angle, ab d. 

133. — Corollary. As the two angles at 5, {Fig. 95,) are right 
angles, and as the semi-circle, c a d^ contains 180 degrees, {Art. 
131,) the measure of two right angles, therefore, is 180 degrees ; 
of one right angle, 90 degrees ; of half r. right angle, 45 ; of 
one-third of a right angle, 30, &c. 

134. — Defio'Ation. In measuring an angie, {Art. 132,) no re- 
gard is to be had to the length of its sides, but only to the degree 
of their inclination. Hence equal angles are such as have the 
same degree of inclination, without regard to the length of their 
sides. 




135.^— Axiom. If two straight lines, parallel to one another, 



m 



AMERICAN HOUSE-CARPENTER. 



9ts a b and c d, {Fig, 97,) stand upon another straight line, as e/, 
the angles, a bf and c d fj are equal ; and the angle, a 6 e, is 
equal to the angle, c d e. 

136. — Definition* If a straight line, as a b, {Fig. 96,) stand 
obliquely upon another straight line, as c d, then one of the an- 
gles, as a b c, is called an obtuse angle, and the other, as ab dj 
an acute angle. 

137. — Axiom. The two angles, a b d and a b c, {Fig. 96,) are 
together equal to two right angles, {Ai^t. 130, 133;) also, the 
three angles, a b d, e b a and c 6 e, are together equal to two right 
angles. 

138. — Corollary. Hence all the angles that can be made upon 
one side of a line, meeting in a point in that line, are together 
equal to two right angles. 

139. — Corollary. Hence all the angles that can be made on 
both sides of a line, at a point in that line, or all the angles that 
can be made about a point, are together equal to four right angles. 




6 d 




Fia 98. 



140. — Proposition. If to each of two equal angles a third 
angle be added, their sums will be equal. Let ab c and d e f, 
{Fig. 98,) be equal angles, and the angle, i j k, the one to be 
added. Make the angles, gb a and hed, each equal to the given 
angle, ij k ; then the angle, g b r, will be equal to the angle, h e 
f; for, if a b c and d e/be angles of 90 degrees, and i j k, 30, 
then the angles, g b c and h ef, will be each equal to 90 and 
30 added, viz : 120 degrees. 



PRACTICAL GEOMETRY. 



66 





Ki-' !)i>. 



141. — Proposition. Triangles that have two of their sides 
and the angle contained between them respectively equal, have 
also their third sides and the two remaining angles equal ; and 
consequently one triangle will every way equal the other. Let a 
b c, {Fig, 99,) and d efhe two given triangles, having the angle 
at a equal to the angle at d, the side, a b, equal to the side, d e, 
and the side, a c, equal to the side, df; then the third side of 
one, b c, is equal to the third side of the other, ef; the angle at b 
is equal to the angle at e, and the angle at c is equal to the angle 
at/. For, if one triangle be applied to the other, the three points, 
b, a, c, coinciding with the three points, e, d, /, the line, b c, must 
coincide with the line, e f; the angle at b with the angle at e ; 
the angle at c with the angle at/; and the triangle, 6 a c, be every 
way equal to the triangle, e df. 




142. — Proposition. The two angles at the base of an isoceles 
triangle are equal. Let ab c, {Fig. 100,) be an isoceles triangle, 
of which the sides, a b and a c, are equal. Bisect the angle, {Art, 



56 



AMERICAN HOUSE-CARPENTER. 



77,) b a Cjhy the line, a d. Then the line, b a, being equal to 
the line, a c ; the line, a d, of the triangle, A, being equal to the 
line, a d, of the triangle, B, being conmion to each ; the angle, b 
a dj being equal to the angle, d a c ; the line, b d, must, accord- 
ing to ArL 141, be equal to the line, dc ; and the angle at b must 
be equal to the angle at c. 




B 




Fi?. Mi. 



143. — Proposition. A diagonal crossing a parallelogram di- 
vides it into two equal triangles. Let abed, [Fig. 101,) be a 
given parallelogram, and 6 c, a line crossing it diagonally. Then, 
as a c is equal to b cZ, and a b to c d, the angle at a to the angle 
at d, the triangle. A, must, according to Art. 141, be equal to the 
triangle, B. 



a 




/, 


A 


^^^ 
^^f 


C /^ 
^^ D 


B 




I'l-. 10- 



144. — Proposition. Let a h c d^ (^Vo-- 102,) be a given pa- 
rallelogram, and 6 c a dingoiial. At any distance between a b and 
c c?, draw e/, parallel to a b ; through the point, g, the intersection 
of the lines, b c and ef, draw h «, parallel to b d. In every paral- 
lelogram thus divided, the parallelogram, J, is equal to the paral- 
lelogram, B. According to xirt. 143, the triangle, a 6 c, is 
equal to the triangle, bed; the triangle, C, to the triangle, D ; 
and E to F ; this being the case, take D and F from the triangle, 
bed. and C and E from the triangle, a b c, and what remains 



PRACTICAL GEOMETRY. 



57 



in one must be equal to what remains in the other ; therefore, the 
parallelogram, A, is equal to the parallelogram, B. 




Fig. 103. 



145. — Proposition. Parallelograms standing upon the same 
base and between the same parallels, are equal. Let ah c d and 
ef cd^ {Fig. 1()3,) be given parallelograms, standing upon the 
same base, c d, and between the same parallels, a f and c d. 
Then, ah and e/ being equal to c c/, are equal to one another: 
h e being added to both a h and e f^ a e equals h f ; the line, a c. 
being equal to h d, and a e to hf, and the angle, c a e, being 
equal, {Art. 135,) to the angle, d h f, the triangle, a e c, must be 
equal, {Art. 141,) to the triangle, hfd; these two triangles being 
equal, take the same amount, the triangle, beg, from each, and 
what remains in one, a h g c, must be equal to what remains in 
the other, efdg; these two quadrangles being equal, add the 
same amount, the triangle, c g d, to each, and they must still be 
equal ; therefore, the parallelogram, ah c d, is equal to the paral- 
lelogram, efcd. 

146. — Corollary. Hence, if a parallelogram and triangle stand 
upon the same base and between the same parallels, the parallelo- 
gram will be equal to double the triangle. Thus, the paral- 
lelogram, a dj {Fig. 103,) is double, {Art. 143,) the triangle, 
c e d. 

147. — Proposition. Let a h c d, {Fig. 104,) be a given quad- 
rangle with the diagonal, a d. From h, draw b e, parallel to a d; 
extend cdto e ; join a and e ; then the triangle, <z e c, will be equal 
in area to the quadrangle, a he d. Since the triangles, adh and 
a d e^ stand upon the same base, a d, and between the same paral- 

8 



58 



AMERICAN HOUSE-CARPENTER. 




Fig. 104. 



lels, a d and h e, they are therefore equal, {^Art. 145, 146 ;) and 
since the triangle, C, is common to both, the remaining triangles, A 
and 5, are therefore equal ; then B being equal to yl, the triangle, 
a e c, is equal to the quadrangle, a h c d. 




Fig. 105. 



148. — Proposition. If two straight lines cut each other, as 
a h and c c?, {Fig. 10.5,) the vertical, or opposite angles, A and 
C, are equal. Thus, a e, standing upon c d., forms the angles, 
jBand C, which together amount, {Art. 137,) to two right angles ; 
in the same manner, the angles, A and 5, form two right angles ; 
since the angles, A and B^ are equal to B and C, take the same 
amount, the angle, J5, from each pair, and what remains of one 
pair is equal to what remains of the other ; therefore, the an- 
gle, A, is equal to the angle, C. The same can be proved of 
the opposite angles, B and D. 

149. — Proposition. The three angles of any triangle are 
equal to two right angles. Let a b c, {Fig'. 106,) be a given tri- 
angle, with its sides extended to/, e, and d, and the line, eg. 



PRACTICAL GEOMETRY. 69 




drawn parallel to h e. As ^ c is parallel to e b, the angle, g c d^ 
is, equal, [Art. 135,) to the angle, e b d ; as the lines, /c and b e, 
cut one another at a, the opposite angles, / a c and b a c, are 
equal, {Art. 14S ;) as the angle, /« c, is equal, [Art. 135,) to the 
angle, a eg, the angle, a c g, is equal to the angle, b a c ; there- 
fore, the three angles meeting at c, are equal to the three angles 
of the triangle, a b c : and since the three angles ate are equal, 
{Art. 137,) to two right angles, the three angles of the triangle, a 
b c, must likewise be equal to two right angles. Any triangle 
can be subjected to the same proof. 

150. — Corollary. Hence, if one angle of a triangle be a right 
angle, the other two angles amount to just one right angle. 

151. — Corollary. If one angle of a triangle be a right angle. 
and the two remaining angles are equal to one another, these are 
each equal to half a right angle. 

152. — Corollary. If any two angles of a triangle amount to 
a right angle, the remaining angle is a right angle. 

153. — Corollary. If any two angles of a triangle are together 
equal to the remaining angle, that remaining angle is a right 
angle. 

154. — Corollary. If any two angles of a triangle are each 
equal to two-thirds of a right angle, the remaining angle is also 
equal to two-thirds of a right angle. 

155. — Corollary. Hence, the angles of an equi-lateral trian- 
gle, are each equal to two-thirds of a right angle. 



60 



AMERICAN HOUSE-CARPENTER. 
6 




\m.~-Proposition. If from the extremities of the diameter of 
a semi-circle, two straight hues be drawn to any point in the cir- 
cumference, the angle formed by them at that point will be a 
right angle. Let a h c, [Fig. 107,) be a given semi-circle ; and 
a 6 and 6 c, lines drawn from the extremities of the diameter, a 
c, to the given point, h ; the angle formed at that point by these 
lines, is a right angle. Join the point, h, and the centre, d; the 
lines, d a,dh and d c, being radii of the same circle, are equal ; 
the angle at a is therefore equal, {Art. 142,) to the angle, a h d, 
also, the angle at c is, for the same reason, equal to the angle, d I 
c; the angle, a h c, being equal to the angles at a and c taken 
together, must therefore, {Art. 153,) be a right angle. 




^^7 .—Proposition. The square of the hypothenuse of a 
right-angled triangle, is equal to the squares of the two remaining 
sides. Let a h c, {Fig. 108,) be a given right-angled triangle, 
having a square formed on each of its sides : then, the square, h e, is 
equal to the squares, A c and ^ 6, taken together. This can be 



PRACTICAL GEOMETRY. 61 

proved by showing that the parallelogram, h l, is equal to the square, 
^b ; and that the parallelogram, c /, is equal to the square, h c. The 
angle, c 6 (/, is a right angle, and the angle, abf, is aright angle ; 
add to each of these the angle, ab c ; then the angle,/ b c, will evi- 
dently be equal, {Art. 140,) to the angle, abd ; the triangle,/ 6 c, 
and the square, g- 6, being both upon the same base,/6, and between 
the same parallels, / b and^ c, the square, g- b. is equal, {Art. 146,) 
to twice the triangle,/ 6 c ; the triangle, a b d, and the parallelo- 
gram, b I, being both upon the sa.me base, b d, and between the 
same parallels, b d and a I, the parallelogram, b Z, is equal to twice 
the triangle, abd; the triangles,/ 6 c and abd, being equal to 
one another, {Art. 141,) the square, g- b, is equal to the parallelo- 
gram, b I, either being equal to twice the triangle, /6 c or a b d. 
The method of proving h c equal to c Z is exactly similar — thus 
proving the square, b e, equal to the squares, h c and ^ 6, taken 
together. 

This problem, which is the 47th of the First Book of Euclid 
is said to have been demonstrated first by Pythagoras. It is sta 
ted, (but the story is of doubtful authority,) that as a thank-oifer 
ing for its discovery he sacrificed a hundred oxen to the gods. 
From this circumstance, it is sometimes called the hecatomb pro- 
blem. It is of great value in the exact sciences, more especially 
in Mensuration and Astionomy, in which many otherwise intri- 
cate calculations are by it made easy of solution. 

These demonstrations, which relate mostly to the problems pre- 
viously given, are introduced to satisfy the learner in regard to 
their mathematical accuracy. By studying and thoroughly un- 
derstanding them, he will soonest arrive at a knowledge of their 
importance, and be likely the longer to retain them in memory. 
Should he have a relish for such exercises, and wish to continue 
them farther, he may consult Euclid's Elements, in which the 
whole subject of theoretical geometry is treated of in a manner 
sufficiently intelligible to be understood by the young mechanic. 



62 AMERICAN HOUSE-CARPENTER. 

The house-carpenter, especially, needs information of this kind, 
and were he thoroughly acquainted with the principles of geome- 
try, he would be much less liable to commit mistakes, and be 
better qualified to excel in the execution of his often difficult un- 
dertakings. 



SKCTlOiN JL— AROHITECTTTRE. 



HISTORY OF ARCHITECTURE. 

158. — Architecture has been defined to be — "the art of build- 
ing ;" but, in its common acceptation, it is — " the art of designing 
and constructing buildings, in accordance with such principles as 
constitute stability, utility and beauty." The literal signification 
of the Greek word archi-tecton^ from which the word architect 
is derived, is chief-carpenter ; but the architect has always been 
known as the chief designer rather than the chief builder. Of 
the three classes into which architecture has been divided — viz., 
Civil, Military, and Naval, the first is that which refers to the 
construction of edifices known as dwellings, churches and other 
public buildings, bridges, (fee, for the accommodation of civilized 
man — and is the subject of the remarks which follow. 

159. — This is one of the most ancient of the arts : the scrip- 
tures inform us of its existence at a very early period. Cain, 
the son of Adam, — " builded a city, and called the name of the 
city after the name of his son, Enoch" — but of the peculiar style 
or manner of building we are not informed. It is presumed that 
it was not remarkable for beauty, but that utility and perhaps sta- 
bility were its characteristics. Soon after the deluge — that me 



64 AMERICAN HOUSE-CARPENTER. 

morable event, which removed from existence all traces of the 
works of man — the Tower of Babel was commenced. This was 
a work of such magnitude that the gathering of the materials, 
according to some writers, occupied three years ; the period from 
its commencement until the work was abandoned, was twenty- 
two years ; and the bricks were like blocks of stone, being twenty 
feet long, fifteen broad and seven thick. Learned men have given 
it as their opinion, that the tower in the temple of Belus at Baby- 
lon was the same as that which in the scriptures is called the 
Tower of Babnl. The tower of the temnle of Belus was square 
at its base, eacn side measuring one iurlong, and consequently 
half a mile in circumference. Its form was that of a pyramid 
and its height was 660 feet. It had a winding passage on the 
outside from the base to the summit, wliich was wide enough for 
two carriages. 

1^0.- -Historical accounts of ancient cities, of which there are 
now but few remains — such as Babylon. Palmyra and Ninevah 
of the Assyrians : Sidon, Tyre, Aradus and Serepta of the Phoe- 
nicians ; and Jerusalem, with its splendid temple, of the Israelites 
— show that architecture among them had made great advances. 
Ancient monuments of the art are found also among other nations j 
the subterraneous temples of the Hindoos upon the islands, Ele- 
phanta and Salsetta ; the ruins of Persepolis in Persia ; pyramids, 
obelisks, temples, palaces and sepulchres in Egypt — all prove that 
the architects of those early times were possessed of skill and 
judgment highly cultivated. The principal characteristics of 
their works, are gigantic dimensions, immoveable solidity, and, in 
some instances, harmonious splendour. The extraordinary size 
of some is illustrated in the pyramids of Egypt. The largest of 
these stands not far from the city of Cairo : its base, which is 
square, covers about 111 acres, and its height is nearly 500 feet. 
The stones of which it is built are immense — the smallest being 
full thirty feet long. 

161. — Among the Greeks, architecture was cultivated as a fine 



ARCHITECTURE. 65 

art, and rapidly advanced towards perfection. Dignity and grace 
were added to stability and magnificence. In the Doric order, 
their first style of building, this is fully exemplified. Phidias, 
Ictinus and Callicrates, are spoken of as masters in the art at this 
period : the encouragement and support of Pericles stimulated 
them to a noble emulation. The beautiful temple of Minerva, 
erected upon the acropolis of Athens, the Propyleum, the Odeum 
and others, were lasting monuments of their success. The Ionic 
and Corinthian orders Avere added to the Doric, and many mag- 
nificent edifices arose. These exemplified, in their chaste propor- 
tions, the elegant refinement of Grecian taste. Improvement in 
Grecian architecture continued to advance, until perfection seems 
to have been attained. The specimens which have been partially 
preserved, exhibit a combination of elegant proportion, dignified 
simplicity and majestic grandeur. Architecture among the 
Greeks was at the height of its glory at the period immediately 
preceding the Peloponnesian war ; after which the art declined. 
An excess of enrichment succeeded its former simple grandeur ; 
yet a st«:ict regularity was maintained amid the profusion of orna- 
ment. After the death of Alexander, 323 B. C, a love of gaudy 
splendour increased : the consequent decline of the art was 
visible, and the Greeks afterwards paid but little attention to the 
science. 

162. — While the Greeks were masters in architecture, which 
they applied mostly to their temples and other public buildings, 
the Romans gave their attention to the science in the construction 
of the many aqueducts and sewers with which Rome abounded ; 
building no such splendid edifices as adorned Athens, Corinth 
and Ephesus, until about 200 years B. C, when their intercourse 
with the Greeks became more extended. Grecian architecture 
was introduced into Rome by Sylla ; by whom, as also by Marius 
and Caesar, many large edifices were erected in various cities of 
Italy. But imder Caesar Augustus, at about the beginning of the 
christian era, the art arose to the greatest perfection it ever at- 

9 



66 AMERICAN HOUSE-CARPENTER. 

tained in Italy. Under his patronage, Grecian artists were en- 
couraged, and many emigrated to Rome. It was at about this 
time that Solomon's temple at Jerusalem was rebuilt by Herod — 
a Roman. This was 46 years in the erection, and was most pro- 
bably of the Grecian style of building — perhaps of the Corin- 
thian order. Some of the stones of which it was built were 46 
feet long, 21 feet high and 14 thick ; and others were of the 
astonishing length of 82 feet, The porch rose to a great height ; 
the whole being built of white marble exquisitely polished. This 
IS the building concerning which it was remarked — " Master, see 
vvhat manner of stones, and what buildings are here." For the 
construction of private habitations also, finished artists were em- 
ployed by the Romans : their dwellings being often built with the 
finest marble, and their villas splendidly adorned. After Augus- 
tus, his successors continued to beautify the city, until the reign of 
Constantine ; who, having removed the imperial residence to 
Constantinople, neglected to add to the splendour of Rome ; and 
the art, in consequence, soon fell from its high excellence. 

Thus we find that Rome was indebted to Greece for what she 
possessed of architecture — not only for the knowledge of its prin- 
ciples, but also for many of the best buildings themselves ; these 
having been originally erected in Greece, and stolen by the un- 
principled conquerors — taken down and removed to Rome. 
Greece was thus robbed of her best monuments of architecture. 
Touched by the Romans, Grecian architecture lost much of its 
elegance and dignity. The Romans, though justly celebrated 
for their scientific knowledge as displayed in the construction of 
their various edifices, were not capable of appreciating the simple 
grandeur, the refined elegance of the Grecian style ; but sought 
to improve upon it by the addition of luxurious enrichment, and 
thus deprived it of true elegance. In the days of Nero, whose 
palace oif gold is so celebrated, buildings were lavishly adorned. 
Adrian did much to encourage the art ; but not satisfied with the 
simplicity of the Grecian style, the artists of his time aimed at 



ARCHITECTURE. 67 

inventing new ones, and added to the already redundant embel- 
lishments of the previous age. Hence the origin of the pedestal, 
the great variety of intricate ornaments, the convex frieze, the 
round and the open pediments, &c. Tlie rage for luxury 
continued until Alexander Severus, who made some improve- 
ment ; but very soon after his reign, the art began rapidly to 
decline, as particularly evidenced in the mean and trifling charac- 
ter of the ornaments. 

163. — The Goths and Yandals, when they overran the coun- 
tries of Italy, Greece, Asia and Africa, destroyed most of the 
works of ancient architecture. Cultivating no art but that of 
war, these savage hordes could not be expected to take any interest 
in the beautiful forms and proportions of their habitations. From 
this time, architecture assumed an entirely different aspect. The 
celebrated styles of Greece were unappreciated and forgotten ; and 
modern architecture took its first step on the platform of existence. 
The Goths, in their conquering invasions, gradually extended it 
over Italy, France, Spain, Portugal and Germany, into England. 
From the reign of Gallienus may be reckoned the total extinction 
of the arts among the Romans. From his time until the 6th or 
7th century, architecture was almost entirely neglected. The 
buildings which were erected during this suspension of the arts, 
were very rude. Being constructed of the fragments of the edi- 
fices which had been demolished by the Visigoths in their unre- 
strained fury, and the builders being destitute of a proper know- 
ledge of architecture, many sad blunders and extensive patch- 
work might have been seen in their construction — entablatures 
inverted, columns standing on their wrong ends, and other ridi- 
culous arrangements characterized their clumsy work. The vast 
number of columns which the ruins around them afforded, they 
used as piers in the construction of arcades — which by some is 
thought, after having passed through various changes, to have 
been the origin of the plan of the Gothic cathedral. Buildings 
generally, which ar3 not of the classical styles, and which were 



68 AMERICAN HOUSE-CARPENTER. 

erected after the fall of the Roman empire, have by some been 
indiscriminately included under the term Gothic. But the 
changes which architecture underwent during the dark ages, show 
that there were several distinct modes of building. 

164. — Theodoric, king of the Ostrogoths, a friend of the arts, 
who reigned in Italy from A. D. 493 to 525, endeavoured to re- 
store and preserve some of the ancient buildings ; and erected 
others, the ruins of which are still seen at Verona and Ravenna. 
Simplicity and strength are the characteristics of the structures 
erected by him ; they are, however, devoid of grandeur and ele- 
gance, or fine proportions. These are properly of the Gothic 
style ; by some called the old Gothic to distinguish it from the 
pointed style, which is generally called modern Gothic. 

165. — The Lombards, who ruled in Italy from A. D. 568, had 
no taste for architecture nor respect for antiquities. Accordingly, 
they pulled down the splendid monuments of classic architecture 
which they found standing, and erected in their stead huge build- 
ings of stone wjiich were greatly destitute of proportion, elegance 
or utility — their characteristics being scarcely anything more than 
stability and immensity combined with ornaments of a puerile cha- 
racter. Their churches were disfigured with rows of small columns 
along the cornice of the pediment, small doors and windows with 
circular heads, roofs supported by arches having arched buttresses 
to resist their thrust, and a lavish display of incongruous orna- 
ments. This khid of architecture is called, the Lombard style, 
and was employed in the 7th century in Pa via, the chief city of 
the Lombards ; at which city, as also at many other places, a 
great many edifices were erected in accordance with its inelegant 
forms. 

166. — The Byzantine architects, from Byzantium, Constantino- 
ple, erected many spacious edifices ; among which are included 
the cathedrals of Bamberg, Worms and Mentz, and the most an 
cient part of the minster at Strasburg ; in all of these they com- 
bined the Roman-Ionic order with the Gothic of the Lombards, 



ARCHITECTURE. 69 

This style is called the Lombard-Byzantine. To the last style 
there were afterwards added cupolas similar to those used in the 
east, together with numerous slender pillars with tasteless capi- 
tals, and the many minarets which are the characteristics of the 
proper Byzantine^ or Oriental style. 

167. — In the eighth century, when the Arabs and Moors de- 
stroyed the kingdom of the Goths, the arts and sciences were 
mostly in possession of the Musselmen-conquerors ; at which 
time there were three kinds of architecture practised ; viz : the 
Arabian, the Moorish and the modern-Gothic. The Arabian 
style was formed from Greek models, having circular arches 
added, and towers which terminated with globes and mir.arets. 
The Moorish is very similar to the Arabian, being distinguished 
from it by arches in the form of a horse-shoe. It originated in 
Spain in the erection of buildings with the ruins of Roman archi- 
tecture, and is seen in all its splendour in the ancient palace of the 
Mohammedan monarchs at Grenada, called the Alhambra, or red- 
house. The Modern-Gothic was originated by the Visigoths 
in Spain by a combination of the Arabian and Moorish styles ; 
and introduced by Charlemagne into Germany. On account of 
the changes and improvements it there underwent, it was, at about 
the 13th or 14th century, termed the German^ or romantic style. 
It is exhibited in great perfection in the towers of the minster of 
Strasburgh, the cathedral of Cologne and other edifices. The 
most remarkable features of this lofty and aspiring style, are the 
lancet or pointed arch, clustered pillars, lofty towers and flying 
buttresses. It was principally employed in ecclesiastical archi- 
tecture, and in this capacity introduced into France, Italy, Spain, 
and England. 

168. — The Gothic architecture of England is divided into the 
Norman^ the Early- English^ the Decorated^ and the Perpen- 
dicular styles. The Norman is principally distinguished by the 
character of its ornaments — the chevron, or zigzags being the 
most common. Buildings in this style were erected in the 12th 



70 AMERICAN HOUSE-CARPENTER. 

century. The Early-English is celebrated for the beauty of its 
edifices, the chaste simplicity and purity of design which they 
display, a-nd the peculiarly graceful character of its foliage. This 
style is of the 18th century. The Decorated style, as its name 
implies, is characterized by a great profusion of enrichment, 
which consists principally of the crocket, or feathered-ornament, 
and ball-flower. It was mostly in use in the 14th century. The 
Perpendicular style, which dates from the 15th century, is distin- 
guished by its high towers, and parapets surmounted with spires 
similar in number and grouping to oriental minarets. 

169. — Thus these several styles, which have been erroneously 
terme Jl Got kic, were distinguished bypeculiar characteristics as well 
as by different names. The first symptoms of a desire to return to a 
pure style in architecture, after the ruin caused by the Goths, was 
manifested in the character of the art as displayed in the church 
of St. Sophia at Constantinople, which was erected by Justinian 
in the 6th century. The church of St. Mark at Yenice, which 
arose in the 10th or 11th century, was the work of Grecian archi- 
tects, and resembles in magnificence the forms of ancient archi- 
tecture. The cathedral at Pisa, a v.^onderful structure for the age, 
was erected by a Grecian architect in 1016. The marble with 
which the walls of this building were faced, and of which the four 
rows of columns that support the roof are composed, is said to be 
of an excellent character. The Campanile, or leaning-tower as it 
is usually called, was erected near the cathedral in the 12th cen- 
tury. Its inclination is generally supposed to have arisen from 
a poor foundation ; although by some it is said to have been thus 
constructed originally, in order to inspire in the minds of the 
beholder sensations ot b;ublimity and awe. In the 13th century, 
the science in Italy was slowly progressing ; many fine churches 
were erected, the style of which displayed a decided advance in 
the progress towards pure classical architecture. In other parts 
of Europe, the Gothic, or pointed style, was prevalent. The 
cathedral at Strasburg, designed by Irwin Steinbeck, was erected 



ARCHITECTURE. 71 

in the 13th and 14th centuries. In France and England during 
the 14th century, many very superior edifices were erected in this 
style. 

170. — In the 14th and 15th centuries, and particularly in the 
latter, architecture in Italy was greatly revived. The masters began 
to study the remains of ancient Roman edifices ; and many splen- 
did buildings were erected, which displayed a purer taste in the 
science. Among others, St. Peter's of Rome, which was built 
about this time, is a lasting monument of the architectural skill of 
the age. Giocondo, Michael Angelo, Palladio, Yignola, and other 
celebrated architects, each in their tm-n, did much to restore the art 
to its former excellence. In the edifices which were erected under 
their direction, however, it is plainly to be seen that they studied 
not from the pure models of Greece, but from the remains of the 
deteriorated architecture of Rome. The high pedestal, the cou- 
pled columns, the rounded pediment, the many curved-and-twisted 
enrichments, and the convex frieze, were unknown to pure Gre- 
cian architecture. Yet their efibrts were serviceable in correcting, 
to a good degree, the very impure taste that had prevailed since 
the overthrow of the Roman empire. 

. 171. — At about this time, the Italian masters and numerous 
artists who had visited Italy for the purpose, spread the Roman 
style over various countries of Europe ; which was gradually re- 
ceived into favor in place of the modern-Gothic. This fell into 
disuse ; although it has of late years been again cultivated. It 
requires a building of great magnitude and complexity for a per- 
fect display of its beauties. In America at the present time, the 
pure Grecian style is more or less studied ; and perhaps the sim- 
plicity of its principles is better adapted to a republican country, 
than the intricacy and extent of those of the Gothic. 

STYLES OF ARCHITECTURE. 

172. — It is generally acknowledged that the various styles m 
architecture, were originated in accordance with the different pur- 



72 AMERICAN HOUSE-CARPENTER. 

suits of the early inhabitants of the earth ; and were brought by 
their descendants to their present state of perfection, through the 
propensity for imitation and desire of emulation which are found 
more or less among all nations. Those that followed agricultural 
pursuits, from being employed constantly upon the same piece of 
land, needed a permanent residence, and the wooden hut was the 
offspring of their wants ; while the shepherd, who followed his 
flocks and was compelled to traverse large tracts of country for 
pasture, found the teiii to be the most portable habitation ; again, 
the man devoted to hunting and fishing — an idle and vagabond 
way of living — is naturally supposed to have been content with 
the caver 71 as a place of shelter. The latter is said to have been 
the origin of the Egyptian style; while the curved roof of Chi- 
nese structures gives a strong indication of their having had the 
tent for their model ; and the simplicity of the original style of 
the Greeks, (the Doric.) shows quite conclusively, as is generally 
conceded, that its original was of wood. The modern-Gothic, or 
pointed style, which was most generally confined to ecclesiastical 
structures, is said by some to have originated in an attempt to 
imitate the bower, or grove of trees, in which the ancients per- 
formed their idol-worship. 

173. — There are numerous styles, or orders, in architecture ; 
and a knowledge of the peculiarities of each, is important to the 
student in the art. The Stylobate is the substructure, or base- 
ment, upon which the columns of an order are arranged. In 
Roman architecture — especially in the interior of an edifice — it 
frequently occurs that each column has a separate substructure ; 
this is called a pedestal. If possible, the pedestal should be 
avoided in all cases ; because it gives to tlie column the appear- 
ance of having been originally designed for a small building, 
and afterwards pieced-out to make it long enough for a larger 
one. 

174. — An Order, in architecture, is composed of tw i princi- 
pal parts, viz : the column and the entablature. 



ARCHITECTURE. 73 

175. — The Column is composed of the base, shaft and capital. 

176. — The Entablature, above and supported by the 
columns, is horizontal ; and is composed of the architrave, frieze 
and cornice. These principal parts are again divided into various 
members and mouldings. (See Sect. III.) 

177. — The Base of a column is so called from basis, a founda- 
tion, or footing. 

178. — The Shaft, the upright part of a column standing upon 
the base and crowned with the capital, is from shaflo, to dig — 
in the manner of a well, whose inside is not unlike the form of a 
column. 

179. — The Capital, from kephale or caput, the head, is the 
uppermost and crowning part of the column. 

180. — The Architrave, from archi, chief or principal, and 
trahs, a beam, is that part of the entablature which lies in imme- 
diate connection with the cokimn. 

181. — The Frieze, from^^iron, a fringe or border, is that part 
of the entablature which is immediately above the architrave and 
beneath the cornice. It was called by some of the ancients, 
zophorus, because it was usually enriched with sculptured 
animals. 

182. — The Cornice, from corona, to crown, is the upper and 
projecting part of the entablature — being also the uppermost and 
crowning part of the whole order. 

183. — The Pediment, above the entablature, is the triangu- 
lar portion which is formed by the inclined edges of the roof at 
the end of the building. In Gothic architecture, the pediment is 
called, a gable. 

184. — The Tympanum is the perpendicular triangular surface 
which is enclosed by the cornice of the pediment. 

185. — The Attic is a small order, consisting of pilasters 
and entablature, raised above a larger order, instead of a pedi- 
ment. An attic story is the upper story, its windows being usually 
square. 

10 



M AMERICAN HOUSE-CARPENTER. 

186. — All order, in architecture, has its several parts and mem- 
bers proportioned to one another by a scale of 60 equal parts, 
which are called minutes. If the height of buildings were al- 
ways the same, the scale of equal parts would be a fixed quan- 
tity — an exact number of feet and inches. But as buildings are 
erected of different heights, the column and its accompaniments 
are required to be of different dimensions. To ascertain the scale 
of equal parts, it is necessary to know the height to which the 
whole order is to be erected. This must be divided by the num- 
ber of diameters which is directed for the order under considera- 
tion. Then the quotient obtained by such division, is the length 
of the scale of equal parts — and is, also, the diameter of the 
column next above the base. For instance, in the Grecian Doric 
order the whole height, including column and entablature, is 8 
diameters. Suppose now it were desirable to construct an exam- 
ple of this order, forty feet high. Then 40 feet divided by 8, 
gives 5 feet for the length of the scale ; and this being divided by 
60, the scale is completed. The upright columns of figures, 
marked ^and P, by the side of the drawings illustrating the orders, 
designate the height and the projection of the members. The 
projection of each member is reckoned from a line passing through 
the axis of the column, and extending above it to the top of the 
entablature. The figures represent minutes, or 60ths, of the 
major diameter of the shaft of the column. 

187. — Grecian Styles. The original method of building 
among the Greeks, was in what is called the Doric order : to 
this were afterwards added the Ionic and the Corinthian, 
These three were the only styles known among them. Each 
is distinguished from the other two, by not only a peculiarity 
of some one or more of its principal parts, but also by a 
particular destination. The character of the Doric is robust, 
manly and Herculean-like ; that of the Ionic is more delicate, 
feminine, matronly ; while that of the Corinthian is extremely 
delicate, youthful and virgin- like. However they may differ in 



ARCHITECTURE. 



75 



their general character, they are alike famous for grace and dig- 
nity, elegance and grandeur, to a high degree of perfection. 

188. — The Doric Order is so ancient that its origin is un- 
known — although some have pretended to have discovered it. 
But the most general opinion is, that it is an improvement upon 
the original log huts of the Grecians. These no doubt were very 
rude, and perhaps not unlike the following figure. 

The trunks of trees, set 



perpendicularly to support 
the roof, may be taken for 
columns ; the tree laid upon 
the tops of the perpendicu- 
lar ones, the architrave ; the 
ends of the cross-beams 




Fig. 109. 



which rest upon the architrave, the triglyphs ; the tree laid on 
the cross-beams as a support for the ends of the rafters, the bed- 
moulding of the cornice ; the ends of the rafters which project 
beyond the bed-moulding, the mutules; and perhaps the projection 
of the roof in front, to screen the entrance from the weather, gave 
origin to the portico. 

The peculiarities of the Doric order are the triglyphs — those 
parts of the frieze which have perpendicular channels cut in their 
surface; the absence of a base to the column — as also of fillets 
between the flutings of the column, and the plainness of the 
capital. The triglyphs are to be so disposed that the width of 
the metopes — the spaces between the triglyphs — shall be equal to 
their height. 

189. — The i7itercohiinniatio7i^ or space between the columns, 
is regulated by placing the centres of the columns under the cen- 
tres of the triglyphs — except at the angle of the building ; where, 
as may be seen in Fig. 110, one edge of the triglyph must be 
over the centre of the column.* Where the columns are so dis- 
posed that one of them stands beneath every other triglyph, the 
arrangement is called, mono-triglyph^ and is most common. 

* See note, page 108. 



DORIC ORDER. 




Fig. 110. 



ARCHITECTURE. 77 

When a column is placed beneath every third triglyph, the ar- 
rangement is called diastyle ; and when beneath every fourth, 
arcp.ostyle. This last style is the worst, and is seldom practised. 

190. — The Doric order is suitable for buildings that are des- 
tined for national purposes, for banking-houses, (fcc. Its appear- 
ance, though massive and grand, is nevertheless rich and grace- 
ful. The Custom-House and the Union Bank, in New -York city, 
are good specimens of this order. 

191. — The Ionic Order. The Doric was for some time the 
only order in use among the Greeks. They gave their attention 
to the cultivation of it, until perfection seems to have been at- 
tained. Their temples were the principal objects upon which 
their skill in the art was displayed ; and as the Doric order seems 
to have been well fitted, by its massive proportions, to represent 
the character of their male deities rather than the female, there 
seems to have been a necessity for another style which should be 
emblematical of feminine graces, and with which they might 
decorate such temples as were dedicated to the goddesses. Hence 
the origin of the Ionic order. This was invented, according to 
historians, by Hermogenes of Alabanda ; and he being a native 
of Caria, then in the possession of the lonians. the order was 
called, the Ionic. 

192. — The distinguishing features of this order are the volutes^ 
or spirals of the capital ; and the dentils among the bed-mould- 
ings of the cornice : although in some instances, dentils are want- 
ing. The volutes are said to have been designed as a represen- 
tation of curls of hair on the head of a matron, of whom the 
whole column is taken as a semblance. 

193. — The intercolumniation of this and the other orders — 
both Roman and Grecian, with the exception of the Doric — are 
distinguished as follows. When the interval is one and a half 
diameters, it is called, pycnostyle, or columns thick-set ; when 
two diameters, systyle ; when two and a quarter diameters, 
eustyle ; when three diameters, diastyle ; and when more than 



78 



IONIC. 




Fig. 111. 



ARCHITECTURE. 



79 



three diameters, arceostyle, or columns thin-set. In all the orders, 
when there are four columns in one row, the arrangement is 
called, tetrastyle ; when there are six in a rov/, hexastyle ; and 
when eight, octastyle. 

194. — The Ionic order is appropriate for churches, colleges, 
seminaries, libraries, all edifices dedicated to literature and the 
arts, and all places of peace and tranquillity. The front of the 
Merchants' Exchange, New- York city, is a good specimen of this 
order. 




Fip. 112. 



SQ 



AMERICAN HOUSE-CARPENTER. 




Fiff. 113. 



195. — To describe the Ionic volute. Draw a perpendicular 
from a to 5, [Fig. 112,) and make a s equal to 20 min. or to f of 
the whole height, a c ; draw s 0, at right angles to s a, and equal 
to 11 min. ; upon 0, with 2, min. for radius, describe the eye of 
the volute ; a!)'wi;t 0, the centre of the eye, draw the square, r 1 1 
2, with sides equal to half the diameter of the eye, viz., 2^ min., 
and divide it into 144 equal parts, as shown at Fig. 113. The 
several centres in rotation are at the angles formed by the heavy 
lines, as figured, 1,2, 3, 4, 5, 6, &c. The position of these an- 
gles is determined by commencing at the point, 1, and making 
each heavy line one part less in length than the preceding one. 
No. 1 is the centre for the arc, a b, (Fig. 112 ;) 2 is the centre for 
the arc, be; and so on to the last. The inside spiral line is to be 
described from the centres, x, x^ x^ (fee, (Fig. 113,) being the 
Cv ! !(' of the first small square towards the middle of the eye 
f r< m the centre for the outside arc. The breadth of the fillet at 
aj^ is to be made equal to 2-j\ min. This is for a spiral of three 
revolutions ; but one of any number of revolutions, as 4 or 6, 



ARCHITECTURE. 



81 



may be drawn, by dividing o/, {Fig- 113,) into h corresponding 
number of equal parts. Then divide the part nearest the centre, 
0, into two parts, as at h ; join o and 1, also o and 2 ; draw h 3, pa- 
rallel to 1, and h 4, parallel to o 2 ; then the lines, o 1, o 2, A 3, A 
4, will determine the length of the heavy lines, and the place of 
the centies. (See Art. 396.) 

196. — The Corinthian Order is in general like the Ionic, 
though the proportions are lighter. The Corinthian displays a 
more airy elegance, a richer appearance ; but its distinguishing 
feature is its beautiful capital. This is generally supposed to have 
had its origin in the capitals of the columns of Egyptian temples ; 
which, though not approaching it in elegance, have yet a similari- 
ty of form with the Corinthian. The oft-repeated story of its 
origin which is told by Yitruvius — an architect who flourished in 
Rome, in the days of Augustus Caesar — though pretty generally 
considered to be fabulous, is nevertheless worthy of being again 
recited. It is this : a young lady of Corinth was sick, and 
finally died. Her nurse gathered into a deep basket, such trinkets 
and keepsakes as the lady had been fond of when alive, and 
placed them upon her grave ; covering the basket with a flat stone 
or tile, that its contents might not be disturbed. The basket was 
placed accidentally upon the stem of an acanthus plant, which, 
shooting forth, enclosed the basket with its foliage ; some of which, 
reaching the tile, turned gracefully over in the form of a volute. 

A celebrated sculptor, Calima 
chus, saw the basket thus decorated, 
and from the hint which it sug- 
gested, conceived and constructed a 
capital for a column. This was 
called Corinthian from the fact that it 
was invented and first made use of 
at Corinth. 

197. — The Corinthian being the gayest, the richest and most 
lovely of all the orders, it is appropriate for edifices which ai 

11 




Fig. 114. 



82 



CORINTHIAN. 




13 

~4 
1 
5 

~4 
6 
5 
5 



24^ 

45 
41 
39 



15 
~5 
15 

3i 



J5 

27 

28* 

25 



33 
35 
35 
33 
38 

43 




mmm 



L 



1 




)_ 



Fig. 115 



ARCHITECTURE. S3 

dedicated to amusement, banqueting and festivity — for ail places 
where delicacy, gayety and splendour are desirable. 

198. — In addition to the three regular orders of architecture, it 
was sometimes customary among the Greeks — and afterwards 
among other nations — to employ representations of the human 
form, instead of columns, to support entablatures ; these were 
called Persians and Caryatides. 

199. — Persians are statues of men, and are so called in com 
memoration of a victory gained over the Persians by Pausanias. 
The Persian prisoners were brought to Athens and condemned to 
abject slavery ; and in order to represent them in the lowest state 
of servitude and degradation, the statues were loaded with the 
heaviest entablature, the Doric. 

200. — Caryatides are statues of women dressed in long robes 
after the Asiatic manner. Their origin is as follows. In a war 
between the Greeks and the Caryans, the latter were totally van- 
quished, their male population extinguished, and their female& 
carried to Athens. To perpetuate the memory of this event, 
statues of females, having the form and dress of the Caryans, were 
erected, and crowned with the Ionic or Corinthian entablature. 
The caryatides were generally formed of about the human size, 
but the Persians much larger ; in order to produce the greater awe 
and astonishment in the beholder. The entablatures were pro- 
portioned to a statue in like manner as to a column of the same 
height. 

201. — These semblances of slavery have been in frequent use 
among moderns as well as ancients : and as a relief from the 
stateliness and formality of the regular orders, are capable ot 
forming a thousand varieties ; yet in a land of liberty such marks 
of human degradation ought not to be perpetuated. 

202. — Roman Styles. Strictly speaking, Rome had no 
architecture of her own — all she possessed was borrowed from 
other nations. Before the Romans exchanged intercourse with 
the Greeks, they possessed some edifices of considerable extent 



g4 AMERICAN HOUSE-CARPENTER. 

and merit, which were erected by architects from Etruria ; but 
Rome was principally indebted to Greece for what she acquired 
of the art. Although there is no such thing as an architecture of 
Roman invention, yet no nation, perhaps, ever was so devoted to 
the cultivation of the art as the Roman. Whether we consider 
the number and extent of their structures, or the lavish richness 
and splendour with which they were adorned, we are compelled 
to yield to them our admiration and praise. At one time, under 
the consuls and emperors, Rome employed 400 architects. The 
public works — such as theatres, circuses, baths, aqueducts, <fec. — 
were, in extent and grandeur, beyond any thing attempted in 
modern times. Aqueducts Avere built to convey water from a 
distance of 60 miles or more. In the prosecution of this work, 
rocks and mountains were tunnelled, and valleys bridged. Some 
of the latter descended 200 feet below the level of the water ; and 
in passing them the canals were supported by an arcade, or suc- 
cession of arches. Public baths are spoken of as large as cities ; 
being fitted up with numerous conveniences for exercise and 
amusement. Their decorations were most splendid ; indeed, the 
exuberance of the ornaments alone was offensive to good taste. 
So overloaded with enrichments were the baths of Diocletian, 
that on an occasion of public festivity, great quantities of sculp- 
ture fell from the ceilings and entablatures, killing many of the 
people. 

203. — The three orders of Greece were introduced into Rome 
in all the richness and elegance of their perfection. But the luxu- 
rious Romans, not satisfied with the simple elegance of their re- 
fined proportions, sought to improve upon them by lavish displays 
of ornament. They transformed in many instances, the true ele- 
gance of the Grecian art into a gaudy splendour, better suited to 
their less refined taste. The Romans remodelled each of the 
orders : the Doric was modified by increasing the height of the 
column to 8 diameters ; by changing the echinus of the capital 
for an ovolo, or quarter-round, and adding an astragal and neck 



ARCHITECTURE. 85 

belvow it ; by placing the centre of the first triglyph, instead of 
one edge, over the centre of the column ; and introducing hori- 
zontal instead of inclined mutules in the cornice. The Ionic 
was modified by diminishing the size of the volutes, and, in some 
specimens, introducing a new capital in which the volutes were 
diagonally arranged. This new capital has been termed modern 
Ionic. The favorite order at Rome and her colonies was the Co- 
rinthian. The Roman artists, in their search for novelty, sub- 
jected it to many alterations — especially in the foliage of its capi- 
tal. Into the upper part of this, they introduced the modified 
Ionic capital ; thus combining the two in one. This change was 
dignified with the importance of an order^ and received the ap- 
pellation Composite, or Roman : the best specimen of which is 
found in the Arch of Titus. This style was not much used 
among the Romans themselves, and is but slightly appreciated 
now. Its decorations are too profuse — a standing monument of 
;he luxury of the age in which it was invented. 

204. — The Tuscan Order is said to have been introduced 
to the Romans by the Etruscan architects, and to have been 
the only style used in Ita'y before the introduction of the 
Grecian orders. However this may be, its similarity to the 
Doric order gives strong indications of its having been a 
rude imitation of that style : this is very probable, since his- 
tory informs us that the Etruscans held intercourse with the 
Greeks at a remote period. The rudeness of this order prevented 
its extensive use in Italy. All that is known concerning it is from 
Vitruvius — no remains of buildings in this style being found 
among ancient ruins. 

205. For mills, factories, markets, barns, stables, &c., where 
utility and strength are of more importance than beauty, the im- 
prove I modification of this order, called the moc^er/i Tuscan, 
[Fig. 116,) will be useful ; and its simplicity recommends it 
where economy is desirable. 

206. — Egyptian Style. The architecture of the ancient 



86 



TUSCAN. 




Fig. 116. 



ARCHITECTURE. 87 

Egyptians— to which tliat of the ancient Hindoos bears some re- 
semblance — is characterized by boldness of outline, solidity and 
grandeur. The imazing labyrinths and extensive artificial lakes, 
the splendid palaces and gloomy cemeteries, the gigantic pyramids 
and towering obelisks, of the Egyptians, were works of immen- 
sity and durability ; and their extensive remains are enduring 
proofs of the enlightened skill of this once-powerful, but long since 
extinct nation. The principal features of the Egyptian Style of 
architecture are — uniformity of plan, never deviating from right 
lines and angles ; thick walls, having the outer surface slightly 
deviating inwardly from the perpendicular ; the whole building 
low ; roof flat, composed of stones reaching in one piece from pier 
to pier, these being supported by enormous columns, very short in 
proportion to their height ; the shaft sometimes polygonal, having 
no base but with a great variety of handsome capitals, the foliage 
of these being of the palm, lotus and other leaves ; entablatures 
having simply an architrave, crowned with a huge cavetto orna- 
mented with sculpture ; and the intercolumniation very narrow, 
usually 11 diameters and seldom exceeding 2|. In the remains 
of a temple, the walls were found to be 24 feet thick ; and at the 
gates of Thebes, the walls at the foundation were 50 feet thick 
and perfectly solid. The immense stones of which these, as well 
as Egyptian walls generally, were built, had both their inside and 
outside surfaces faced, and the joints throughout the body of the 
wall as perfectly close as upon the outer surface. For this reason, 
as well as that the buildings generally partake of the pyramidal 
form, arise their great solidity and durability. The dimensions 
and extent of the buildings may be judged from the temple of 
Jupiter at Thebes, which was 1400 feet long and 300 feet wide — 
exclusive of the porticos, of which there was a great number. 

It is estimated by Mr. Gliddon, U. S. consul in Egypt, that not 
less than 25,000,000 tons of hewn stone wert employed in the 
erection of the Pyramids of Memphis alone, — or enough to con- 
struct 3,000 Bunker-Hill monuments. Some of the blocks are 40 



88 



EGYPTIAN. 



HP. 




I-'ii:. 117. 



ARCHITECTURE. 89 

feet long, and polished with emery to a surprising degree. It is 
conjectured that the stone for these pyramids was brought, by 
rafts and canals, from a distance of 6 or 7 hundred miles. 

207. — The general appearance of the Egyptian style of archi- 
tecture is that of solemn grandeur — amounting sometimes to 
sepulchral gloom. For this reason it is appropriate for cemete- 
ries, prisons, (fee. ; and being adopted for these purposes, it is 
gradually gaining favour. 

A great dissimilarity exists in the proportion, form and general 
features of Egyptian columns. In some instances, there is no 
uniformity even in those of the same building, each differing 
from the others either in its shaft or capital. For practical use 
in this country, Fig. 117 may be taken as a standard of this 
style. The Halls of Justice in Centre-street, New- York city, is 
a building in general accordance with the principles of Egyptian 
architecture. 

Buildings in General. 

208. — That style of architecture is to be preferred in whicn 
utility, stability and regularity, are gracefully blended with gran- 
deur and elegance. But as an arrangement designed for a warm 
country would be inappropriate for a colder climate, it would seem 
that the style of building ought to be modified to suit the wants 
of the people for whom it is designed. High roofs to resist the 
pressure of heavy snows, and arrangements for artificial heat, are 
indispensable in northern climes ; while they would be regarded 
as entirely out of place in buildings at the equator. 

209. — Among the Greeks, architecture was employed chiefly 
upon their temples and other large buildings ; and the proportions 
of the orders, as determined by them, when executed to such 
large dimensions, have the happiest effect. But when used for 
small buildingSjporticos, porches, <fec., especially in country-places, 
they are rather heavy and clumsy ; in such cases, more slender 
proportions will be found to produce a better effect. The 

12 



90 AMERICAN HOUSE-CARPENTER. 

English cottage-style is rather more appropriate, and is becom- 
ing extensively practised for small buildings in the country. 

210. — Every building should bear an expression suited to its 
destination. If it be intended for national purposes, it should be 
magnificent — grand ; for a private residence, neat and modest .; 
for a banqueting-house, gay and splendid ; for a monument or 
cemetery, gloomy — melancholy ; or, if for a church, majestic and 
graceful. By some it has been said — "somewhat dark and 
gloomy, as being favourable to a devotional state of feeling ;" but 
such impressions can only result from a misapprehension of the 
nature of true devotion. " Her ways are ways of pleasantness^ 
and all her paths are peace." The church should rather be a type 
of that brighter world to which it leads. 

211. — However happily the several parts of an edifice may be 
disposed, and however pleasing it may appear as a whole, yet 
much depends upon its site^ as also upon the character and style 
of the structures in its immediate vicinity, and the degree of cul- 
tivation of the adjacent country. A splendid country-seat should 
have the out-houses and fences in the same style with itself, the 
trees and shrubbery neatly trimmed, and the grounds well cul- 
tivated. 

212. — Europeans express surprise that so many houses in this 
country are built of wood. And yet, in a new country, where 
wood is plenty, that this should be so is no cause for wonder. 
Still, the practice should not be encouraged. Buildings erected 
with brick or stone are far preferable to those of wood ; they are 
more durable ; not so liable to injury by fire, nor to need repairs ; 
and will be found in the end quite as economical. A wooden 
house is suitable for a temporary residence only ; and those who 
would bequeath a dwelling to their children, will endeavour to 
build with a more durable material. Wooden cornices and gut- 
ters, attached to brick houses, are objectionable — not only on ac- 
count of their frail nature, but also because they render the build- 
ing liable to destruction by fire. 



91 




Fig. 118 



Fig. 119. 



92 AMERICAN HOUSE-CARPENTER. 

213. — Dwelling houses are built of various dimensions and 
styles, according to their destination ; and to give designs and di- 
rections for their erection, it is necessary to know their situation 
and object. A dwelling intended for a gardener, would require 
very diiferent dimensions and arrangements from one intended for 
a retired gentleman — with his servants, horses, &c. ; nor would 
a house designed for the city, be appropriate for the country. For 
city houses, arrangements that would be convenient for one fa- 
mily, might be very inconvenient for two or more. Fig. 118, 119, 
120 and 121, represent the icJuio graphical projection^ or ground- 
plan, of the floors of an ordinary city house, designed to be occupied 
by one family only. Fig. 122 is an elevation, or front-view, of 
the same house : all these plans are drawn at the same scale — 
which is that at the bottom of Fig. 122. 

Fig. 118 is a plan of the basement. 

a is the dining-room. 

b — kitchen. 

c — wash-room. 

d, d, d, — wash-troughs. 

c, e, — pantries with shelving. 

/ — passage having shelves, drawers, &c.j on one side, and 
clothes-hooks on the other. 
g — kitchen-dresser. 
A, ^, — front and rear areas. 

Fig. 119 — plan of the first-story, 
j^i— parlours. 
k — library. 
I — portico. 

Fig. 120 — plan of the second-story. 
a — toilet and sitting room. 
b — principal bed-chamber. 
c — bath-room. 

d, d, — bed-chambers. 

e — passage with wardrobe and clothes-hooks. 



93 



~^ 



- ». ' 



J=^. 



r 



=n c 



rh c 



/M 



^^=^ ?=^ i 



I f I 



■i= i^ v-^ — "- ^ * 



/ 



'» r 



1 




ji 



s 



7. 



l-T 



a 



h ±h i;. 1 



I \ ' 



7i 



g 



r '1 



.1 1 . 



Ti-. IJU. 



Fijj. 121. 



94 AMERICAN HOUSE-CARPENTER. 

Fig. 121 — plan of the attic-story. 

/—nursery, 

g^ g, gy ^ — bed-chamber«= 

hj A, h, h, /?, — wardrobes 

I — pantry with shelves, 

; — step-ladder leading to roof. 

Fig. 122 — front elevation. 

a — section, 

b — front, 

These are introduced to give some general ideas of the princi- 
ples to be followed in designing city houses. The width of city 
lots is ordinarily 25 feet, but as it has become a common practice 
to reduce this size, on account of the enhanced value of land, the 
plans here given are designed for a lot only 20 feet wide — the or- 
dinary width of many buildings of this class. In placing the 
chimneys, make the parlours of equal size, and set the chimney- 
breast in the middle of the space between the sliding-door parti- 
tion and the front (and rear) v/alls. The basement chimney- 
breasts may be placed in the middle of the side of the room, as 
there is but one flue to pass through the chimney-breast above ; 
but in the second-story, as there are two flues, one from the base- 
ment and one from the parlour, the breast will have to be placed 
nearly perpendicular over the parlour breast, so as to receive the 
flues within the jambs of the fire-place. As it is desirable to 
have the chimney-breast as near the middle of the room as pos- 
sible, it may be placed a few inches towards that point from over 
the breast below. So in arranging those of the stories above, 
always make provision for the flues from below. 

214. — In placing the stairs, there should be at least as much 
room in the passage at the side of the stairs, as upon them ; and in 
regard to the length of the passage in the second story, there must 
be room for the doors which open from each of the principal rooms 
into the hall, and more if the stairs require it. Having assigned 
a position for the stairs of the second story, let the Avinders of 



96 






Ee 



. _— r i ^ ) i 


L 


n 


L i 




i c 




~n 




1 

[L_ 


- 






— 


J 


T 







mil III! 
3 43210 



10 



15 



2fj/Y 



Fig. 122. 



96 AMERICAN HOUSP>CARPENTER. 

tlie Other stories be placed perpendicularly over and under them ; 
and be careful to provide for head-room. To ascertain this, when 
it is doubtful, it is well to draw a vertical section of the whole 
stairs ; but in ordinary cases, this is not necessary. To dispose 
the windows properly, the middle Avindow of each story should 
be exactly in the middle of the front ; but the pier between the 
two windows which light the parlour, should be in the centre of 
that room ; because when chandeliers or any similar ornaments, 
hang from the centre-pieces of the parlour ceilings, it is important, 
ill order to give the better effect, that the pier-glasses at the front 
and rear, be in a range with them. If both these objects cannot 
be attained, an approximation to each must be attempted. The 
piers should in no case be less in width than the window open- 
ings, else the blinds or shutters when thrown open will interfere 
with one another; in general practice, it is well to make the out- 
side piers f of the width of one of the middle piers. When this 
is desirable, deduct the amount of the three openings from the 
width of the front, and the remainder will be the amount of the 
width of all tlie piers ; divide this by 10, and the product will be 
-^- of a middle pier; and then, if the parlour arrangements do not 
interfere, give twice this amount to each corner pier, and three 
times the same amount to each of the middle piers. 

PRINCIPLES OF ARCHITECTURE. 

215. — In the construction of the first habitations of men, frail 
and rude as they must have been, the first and principal object 
was, doubtless, utility — a mere shelter from sun and rain. But 
as successive storms shattered the poor tenement, man was taught 
by experience the necessity of building with an idea to durability. 
And when in his walks abroad, the symmetry, proportion and 
beauty of nature met his admiring gaze, contrasting so strangely 
with the misshapen and disproportioned work of his own hands, 
he was led to make gradual changes ; till his abode was rendered 



ARCHITECTURE. 97 

not only commodious and durable, but pleasant in its appearance ; 
and building became a fine-art, having utility for its basis. 

2] 6. — In all designs for buildings of importance, utility, dura- 
bility and beauty, the first great principles of architecture, should 
be pre-eminent. In order that the edifice be useful, commodious 
and comfortable, the arrangement of the apartments should be 
such as to fit them for their several destinations ; for pubUc as- 
semblies, oratory, state, visitors, retiring, eating, reading, sleeping, 
bathing, dressing, (fee. — these should each have its own peculiar 
form and situation. To accomplish this, and at the same time to 
make their relative situation agreeable and pleasant, producing 
regularity and harmony, require in some instances much skill and 
sound judgment. Convenience and regularity are very import- 
ant, and each should have due attention ; yet when both cannot 
be obtained, the latter should in most cases give place to the for- 
mer. A building that is neither convenient nor regular, whatever 
other good qualities it may possess, will be sure of disappro- 
bation. 

217. — The utmost importance should be attached to such ar- 
rangements as are calculated to promote health : among these, ven- 
tilation is by no means the least. For this purpose, the ceilings of 
the apartments should have a respectable height ; and the sky- 
light, or any part of the roof that can be made moveable, should 
be arranged with cord and puUies, so as to be easily raised and 
lowered. Small openings near the ceiling, that may be closed at 
pleasure, should be made in the partitions that separate the rooms 
from the passages — especially for those rooms which are used for 
sleeping apartments. All the apartments should be so arranged 
as to secure their being easily kept dry and clean. In dwellings, 
suitable apartments should be fitted up for bathings with all the 
necessary apparatus for conveying the water. 

218. — To insure stability in an edifice, it should be designed 
upon well-known geometrical principles : such as science has de- 
monstrated to be necessary and sufficient for firmness and dura- 

13 



P8 AMERICAN HOUSE-CARPENTER. 

bility. It is well, also, that it have the appearance of stability as 
well as the reality ; for should it seem tottering and unsafe, the 
sensation of fear, rather than those of admiration and pleasure, 
will be excited in the beholder. To secure certainty and accu- 
racy in the application of those principles, a knowledge of the 
strength and other properties of the materials used, is indispensa- 
ble ; and in order that the whole design be so made as to be 
capable of execution, a practical knowledge of the requisite 
mechanical operations is quite important. 

219. — The elegance of an architectural design, although chiefly 
depending upon a just proportion and harmony of the parts, will 
be promoted by the introduction of ornaments — provided this be 
judiciously performed. For enrichments should not only be of a 
proper character to suit the style of the building, but should also 
have their true position, and be bestowed in proper quantity. The 
most common fault, and one which is prominent in Roman archi- 
tecture, is an excess of em-ichment : an error which is carefully 
to be guarded against. But those who take the Grecian models 
for their standard, will not be liable to go to that extreme. In 
ornamenting a cornice, or any other assemblage of mouldings, at 
least every alternate member should be left plain ; and those that 
are near the eye should be more finished than those which are dis- 
tant. Although the characteristics of good architecture are utili- 
ty and elegance, in connection with durability, yet some buildings 
are designed expressly for use, and others again for ornament : in 
the former, utility, and in the latter, beauty, should be the gov- 
erning principle. 

220. — The builder should be intimately acquainted with the 
principles upon which the essential, elementary parts of a build- 
ing are founded. A scientific knowledge of these will insure 
certainty and security, and enable the mechanic to erect the most 
extensive and lofty edifices with confidence. The more important 
parts are the foundation, the column, the wall, the lintel, the arch, 
the vault, the dome and the roof. A separate description of the 



ARCHITECTURE. 99 

peculiarities of each, would seem to be necessary ; and cannot 
perhaps be better expressed than in the following language of a 
modern writer on this subject. 

221. — "In laying the Foundation of any building, it is ne- 
cessary to dig to a certain depth in the earth, to secure a solid 
basis, below the reach of frost and common accidents. The 
most solid basis is rock, or gravel which has not been moved. 
Next to these are clay and sand, provided no other excavations 
have been made in the immediate neighbourhood. From this 
basis a stone wall is carried up to the surface of the ground, and 
constitutes the foundation. Where it is intended that the super- 
structure shall press unequally, as at its piers, chimneys, or 
columns, it is sometimes of use to occupy the space betv/een the 
points of pressure by an inverted arch. This distributes the 
pressure equally, and prevents the foundation from springing be- 
tween the different points. In loose or muddy situations, it is 
always unsafe to build, unless we can reach the solid bottoiu 
below. In salt marshes and flats, this is done by depositing tim- 
bers, or driving wooden piles into the earth, and raising walls 
upon them. The preservative quality of the salt will keep these 
timbers unimpaired for a great length of time, and makes the 
foundation equally secure with one of brick or stone. 

222. — The simplest member in any building, though by no 
means an essential one to all, is the Column, ox pillar. This is 
a perpendicular part, commonly of equal breadth and thickness, 
not intended for the purpose of enclosure, but simply for tlie sup- 
port of some part of the superstructure. The principal force 
which a column has to resist, is that of perpendicular pressure. 
In its shape, the shaft of a column should not be exactly cylin- 
drical, but, since the lower part must support the weight of the 
superior part, in addition to the weight Avhich presses equally on 
the whole column, the thickness should gradually decrease from 
bottom to top. The outline of columns should be a little curved, 
so as to represent a portion of a very long spheroid, or paraboloid, 



100 AMERICAN HOUSE-CARPENTER. 

rather than of a cone. This figure is the joint result of two cal- 
culations, independent of beauty of appearance, " One of these 
is, that the form best adapted for stability of base is that of a 
cone; the other is, that the figure, which would be of equal 
strength throughout for supporting a superincumbent weight, 
would be generated by the revolution of two parabolas round the 
axis of the column, the vertices of the curves being at its ex- 
tremities. The swell of the shafts of columns was called the en- 
tasis by the ancients. It has been lately found, that the columns 
of the Parthenon, at Athens, which have been commonly sup- 
posed straight, deviate about an inch from a straight line, and 
that their greatest swell is at about one third of their height. 
Columns in the antique orders are usually made to diminish one 
sixth or one seventh of their diameter, and sometimes even one 
fourth. The Gothic pillar is commonly of equal thickness 
throughout. 

223. — The Wall, another elementary part of a building, may 
be considered as the lateral continuation of the column, answer- 
ing the purpose both of enclosure and support. A wall must 
diminish as it rises, for the same reasons, and in the same propor- 
tion, as the column. It must diminish still more rapidly if it ex- 
tends through several stories, supporting weights at different 
heights. A wall, to possess the greatest strength, must also con- 
sist of pieces, the upper and lower surfaces of which are horizon- 
tal and regular, not rounded nor oblique. The walls of most of 
the ancient structures which have stood to the present time, are 
constructed in this manner, and frequently have their stones bound 
together with bolts and cramps of iron. The same method is 
adopted in such modern structures as are intended to possess great 
strength and durability, and, in some cases, the stones are even 
dove-tailed together, as in the light-houses at Eddystone and Bell 
Rock, But many of our modern stone walls, for the sake of 
cheapness, have only one face of the stones squared, the inner 
^alf of the wall being completed with brick ; so that they can, 



ARCHITECTURE. 101. 

in reality, be considered only as brick walls faced with stone. 
Such walls are said to be liable to become convex outwardly, from 
the difference in the shrinking of the cement. Rubble walls are 
made of rough, irregular stones, laid in mortar. The stones 
should be broken, if possible, so as to produce horizontal surfaces. 
The coffer walls of the ancient Romans were made by enclosing 
successive portions of the intended wall in a box, and filling it 
with stones, sand, and mortar, promiscuously. This kind of 
structure must have been extremely insecure. The Pantheon, 
and various other Roman buildings, are surrounded with a double 
brick wall, having its vacancy filled up with loose bricks and 
cement. The whole has gradually consolidated into a mass of 
great firmness. 

The reticulated walls of the Romans, having bricks with 
oblique surfaces, would, at the present day, be thought highly 
unphilosophical. Indeed, they could not long have stood, had it 
not been for the great strength of their cement. Modern brick 
walls are laid with great precision, and depend for firmness more 
upon their position than upon the strength of their cement. The 
bricks being laid in horizontal courses, and continually overlaying 
each other, or breaking joints^ the whole mass is strongly inter- 
woven, and bound together. Wooden walls, composed of timbers 
covered with boards, are a common, but more perishable kind. 
They require to be constantly covered with a coating of a foreign 
substance, as paint or plaster, to preserve them from spontaneous 
decomposition. In some parts of France, and elsewhere, a kind 
of wall is made of earth, rendered compact by ramming it in 
moulds or cases. This method is called building in pise^ and is 
much more durable than the nature of the material would lead 
us to suppose. Walls of all kinds are greatly strengthened by 
angles and curves, also by projections, such as pilasters, chimneys 
and buttresses. These projections serve to increase the breadth 
of the foundation, and are always to be made use of in large 
buildings, and in walls of considerable length. 



102 AMERICAN HOUSE-CARPENTER. 

224. — The Lintel, or heam^ extends in a right line over a 
vacant space, from one column or wall to another. The strength 
of the lintel will be greater in proportion as its transverse vertical 
diameter exceeds the horizontal, the strength being always as the 
square of the depth. The floor is the lateral continuation or 
connection of beams by means of a covering of boards. 

225. — The Arch is a transverse member of a building, an- 
swering the same purpose as the lintel, but vastly exceeding it in 
strength. The arch, unlike the lintel, may consist of any num- 
ber of constituent pieces, without impairing its strength. It is, 
however, necessary that all the pieces should possess a uniform 
shape, — the shape of a portion of a wedge, — and that the joints, 
formed by the contact of their surfaces, should point towards a 
common centre. In this case, no one portion of the arch can be 
displaced or forced inward ; and the arch cannot be broken by 
any force which is not sufficient to crush the materials of which 
it is made. In arches made of common bricks, the sides of which 
are parallel, any one of the bricks might be forced inward, were 
it not for the adhesion of the cement. Any two of the bricks, 
however, by the disposition of their mortar, cannot collective- 
ly be forced inward. An arch of the proper form, when com- 
plete, is rendered stronger, instead of weaker, by the pressure of 
a considerable weight, provided this pressure be uniform. While 
building, however, it requires to be supported by a centring of 
the shape of its internal surface, until it is complete. The upper 
stone of an arch is called the key-stone, but is not more essential 
than any other. In regard to the shape of the arch, its most 
simple form is that of tlie semi-circle. It is, however, very fre- 
quently a smaller arc of a circle, and, still more frequently, a por- 
tion of an ellipse. The simplest theory of an arch supporting 
itself only, is that of Dr. Hooke. The arch, when it has only 
its own weight to bear, may be considered as the inversion of a 
chain, suspended at each end. The chain hangs in such a form, 
that the weight of each link or portion is held in equilibrium by 



ARCHITECTURE. 103 

the result of two forces acting at its extremities ; and these forces, 
or tensions, are produced, the one by the weight of the portion of 
the chain below the Unk, the other by the same weight increased 
by that of the link itself, both of them acting originally in a ver- 
tical direction. Now, supposing the chain inverted, so as to con- 
stitute an arch of the same form and weight, the relative situa- 
tions of the forces will be the same, only they will act in contrary 
directions, so that they are compounded in a similar manner, and 
balance each other on the same conditions. 

The arch thus formed is denominated a catenary arch. In 
common cases, it differs but little from a circular arch of the extent 
of about one third of a whole circle, and rising from the abut- 
ments with an obliquity of about 30 degrees from a perpendicu- 
lar. But though the catenary arch is the best form for support- 
ing its own weight, and also all additional weight which presses 
in a vertical direction, it is not the best form to resist lateral 
pressure, or pressure like that of fluids, acting equally in all direc- 
tions. Thus the arches of bridges and similar structures, when 
covered with loose stones and earth, are pressed sideways, as well 
as vertically, in the same manner as if they supported a Aveight 
of fluid. In this case, it is necessary that the arch should arise 
more perpendicularly from the abutment, and that its general 
figure should be that of the longitudinal segment of an ellipse. 
In small arches, in common buildings, where the disturbing 
force is not great, it is of little consequence what is the shape of 
the curve. The outlines may even be perfectly straight, as in the 
tier of bricks which we frequently see over a window. This is, 
strictly speaking, a real arch, provided the surfaces of the bricks 
tend towards a common centre. It is the weakest kind of arch, 
and a part of it is necessarily superfluous, since no greater portion 
can act in supporting a weight above it, than can be included be- 
tween two curved or arched lines. 

Besides the arches already mentioned, various others are in use. 
The acute or lancet arch, much used in Gothic architecture, is 



104 AMERICAN HOUSE-CARPENTER. 

described usually from two centres outside the arch. It is a 
strong arch for supporting vertical pressure. The rampant arch 
is one in which the two ends spring from unequal heights. The 
horse-shoe or Moorish arch is described from one or more centres 
placed above the base line. In this arch, the lower parts are in 
danger of being forced inward. The ogee arch is concavo-con- 
vex, and therefore fit only for ornament. In describing arches, 
the upper surface is called the extrados^ and llie inner, the in- 
trados. The springing lines are those where the intrados meets 
the abutments, or supporting walls. The span is the distance 
from one springing line to the other. The wedge-shaped stones, 
which form an arch, are sometimes called voiissoirs, the upper- 
most being the key-stone. The part of a pier from which an 
arch springs is called the impost^ and the curve formed by the 
upper side of the voussoirs, the archivolt. It is necessary that 
the walls, abutments and piers, on which arches are supported, 
should be so firm as to resist the lateral thrust^ as well as vertical 
pressure, of the arch. It will at once be seen, that the lateral or 
side way pressure of an arch is very considerable, when we recol- 
lect that every stone, or portion of the arch, is a wedge, a part of 
whose force acts to separate the abutments. For want of atten- 
tion to this circumstance, important mistakes have been committed, 
the strength of buildings materially impaired, and their ruin ac- 
celerated. In some cases, the w^ant of lateral firmness in the 
walls is compensated by a bar of iron stretched across the span of 
the arch, and connecting the abutments, like the tie-beam of a 
roof. This is the case in the cathedral of Milan and some other 
Gothic buildings. 

In an arcade, or continuation of arches, it is only necessary that 
the outer supports of the terminal arches should be strong enough 
to resist horizontal pressure. In the intermediate arches, the lat- 
eral force of each arch is counteracted by the opposing lateral 
force of the one contiguous to it. In bridges, however, where 
individual arches are liable to be destroyed by accident, it is desi- 



ARCHITECTURE. 105 

rable that each of the piers should possess sufficient horizontal 
strength to resist the lateral pressure of the adjoining arches. 

226. — The Vault is the lateral continuation of an arch, serving 
to cover an area or passage, and bearing the same relation to the 
arch that the wall does to the column. A simple vault is con- 
structed on the principles of the arch, and distributes its pressure 
equally along the walls or abutments. A complex or groined 
vault is made by two vaults intersecting each other, in whicji 
case the pressure is thrown upon springing points, and is greatly 
increased at those points. The groined vault is common in 
Gothic architecture. 

227. — The Dome, sometimes called cupola^ is a concave cover- 
ing to a building, or part of it, and may be either a segment of a 
sphere, of a spheroid, or of any similar figure. When built of 
stone, it is a very strong kind of structure, even more so than the 
arch, since the tendency of each part to fall is counteracted, not 
only by those above and below it, but also by those on each side. 
It is only necessary that the constituent pieces should have a 
common form, and that this form should be somewliat iilve ihe 
frustum of a pyramid, so that, when placed in its situation, its 
four angles may point toward the centre, or axis, of the dome. 
During the erection of a dome, it is not necessary that it should 
be supported by a centring, until complete, as is done in the arch. 
Each circle of stones, when laid, is capable of supporting itself 
without aid from those above it. It follows that the dome may 
be left open at top, without a key-stone, and yet be perfectly 
secure in this respect, being the reverse of the arch. The dome 
of the Pantheon, at Rome, has been alwa^^s open at top, and yet 
has stood unimpaired for nearly 2000 years. The upper circle 
of stones, though apparently the weakest, is nevertheless often 
made to support the additional weight of a lantern or tower above 
it. In several of the largest cathedrals, there are two domes, one 
within the other, which contribute their joint support to the lan- 
tern, which rests upon the top. In these buildings, the dome 

14 



106 AMERICAN HOUSE-CARPENTER. 

rests upon a circular wall, which is supported, in its turn, by 
arches upon massive pillars or piers. This construction is called 
•building upon pendentives^ and gives open space and room for 
passage beneath the dome. The remarks which have been made 
ill regard to the abutments of the arch, apply equally to the walls 
immediately supporting a dome. They must be of sufficient 
thickness and solidity to resist the lateral pressure of the dome, 
which is very great. The walls of the Roman Pantheon are of 
^leat depth and solidity. In order that a dome in itself should be 
perfectly secure, its lower parts must not be too nearly vertical, 
since, in this case, they partake of the nature of perpendicular 
vrails, and are acted upon by the spreading force of the parts above 
them. The dome of St. Paul's church, in London, and some 
others of similar construction, are bound with chains or hoops of 
iron, to prevent them from spreading at bottom. Domes which 
are made of wood depend, in part, for their strength, on their in- 
ternal carpentry. The Halle du Bled, in Paris, had originally a 
wooden dome more than 200 feet in diameter, and only one foot 
in thickness. This has since been replaced by a dome of iron. 
(See Art. 303.) 

228. — The Roof is the most common and cheap method of 
covering buildings, to protect them from rain and other effects of 
the weather. It is sometimes fiat, but more frequently oblique, in 
its shape. The flat or platform-roof is the least advantageous for 
shedding rain, and is seldom used in northern countries. The 
pent roof, consisting of two oblique sides meeting at top, is the 
most common form. These roofs are made steepest in cold cli- 
mates, where they are liable to be loaded with snow. Where the 
four sides of the roof are all oblique, it is denominated a hipped 
roof, and where there are two portions to the roof, of different ob- 
liquity, it is a curb, or ^nansard roof. In modern times, roofs 
are made almost exclusively of Avood, though frequently covered 
with incombustible materials. The internal structure or carpen- 
try of roofs is a subject of considerable mechanical contrivance. 



ARCHITECTURE. 107 

The roof is supported by rafters^ which abut on the walls on 
each side, Uke the extremities of an arch. If no other timbers 
existed, except the rafters, they would exert a strong lateral pres- 
sure on the walls, tending to separate and overthrow them. To 
counteract this lateral force, a tie-heam^ as it is called, extends 
across, receiving the ends of tlic rafters, and protecting the wall 
from their horizontal thrust. To prevent the tie-beam from 
sagging^ or bending dovv^nward with its own weight, a king- 
post is erected from this beam, to the upper angle of the rafters, 
serving to connect the whole, and to suspend the weight of the 
beam. This is called tritssing. Qiueeii-posts aie : omctimcs 
added, parallel to the king-post, in iarge roofs ; also various other 
connecting timbers. In Gothic buildings, where the vauUs do 
not admit of the use of a tie-beam, the rafters are prevented from 
spreading, as in an arch, by the strength of the buttresses. 

In comparing the lateral pressure of a high roof with that of a 
low one, the length of the tie-beam being the same, it will be 
seen that a high roof, from its containing most materials, may 
produce the greatest pressure, as far as weight is concerned. On 
the other hand, if the weight of both be equal, then the low roof 
will exert the greater pressure ; and this will increase in propor- 
tion to the distance of the point at which perpendiculars, drawn 
from the end of each rafter, would meet. In roofs, as well as in 
wooden domes and bridges, the materials are subjected to an in- 
ternal strain, to resist which, the cohesive strength of the niateria! 
is relied on. On this account, beams should, when possible, be 
of one piece. Where this cannot be effected, two or more beams 
are connected together by splicing. Spliced beams are never so 
strong as whole ones, yet they may be made to approach the same 
strength, by affixing lateral pieces, or by making the ends overlay 
each other, and connecting them with bolts and straps of iron. 
The tendency to separate is also resisted, by letting the two pieces 
into each other by the process called scarfing. Mortices, in- 



108 AMERICAN HOUSE-CARPENTER. 

tended to truss or suspend one piece by another, should be formed 
"ipon similar principles. 

Roofs in the United States, after being boarded, receive a se- 
condary covering of shingles. When intended to be incombustible, 
they are covered with slates or earth ern tiles, or with sheets of lead, 
copper or tinned iron. Slates are preferable to tiles, being lighter, 
and absorbing less moisture. Metallic sheets are chiefly used for 
flat roofs, wooden domes, and curved and angular surfaces, which 
require a flexible material to cover them, or have not a sufficient 
pitch to shed the rain from slates or shingles. Various artificial 
compositions are occasionally used to cover roofs, the most com- 
mon of which are mixtures of tar with lime, and sometimes with 
sand and gravel." — Ency. Am. (See Art. 285.) 



NOTE TO ARTICLE 189. 



Gi'ECiAN DoiJic Orper. When the xcidth to be occupied by the whole front is limited ; to deter- 
mine tiie dianutLr of ihe colmnn. 
The relation between the parts may be expressed thus : 

_ 00 o 

'^ ~ «:(i + c) + (60 — c) 

Where a eq'i;il« tii • width in feet occupied by the cohTmns, and their intercolumuiations taken 
collectively. ni.'.isi'jvJ at tlie base ; b eci'.ials the width of the metope, in miniUes; c equ;tls the width 
of the trigiypij-- in iuiuuteij; d equals the number of metopes, iX'-A z equals the di;iinet(=r in feet. 

ExampU. — A ir.mi. of six columns— hexiistyle— 61 feet wide; thi- frieze having one triyjlyph over 
each intercolumtiirnion, or mono-triij!y;)h. In this case, there hexw.; five intercoluminatious aud two 
metopes over e:.c!). tiierefore there ai-o 5 X 2=^ 10 metope:?. Let ihe metispe equal 4'2 minutes and 
(he triglyph op;;.; 2-^. T:ien = 01; Z' = 42; r = W; auJ d=z 10; and the formula above becoiaos, 

' - iTi-^-jr^— ^ (GiT^ITo^ - i0^;) + 32= 732"= ^ *"^' ^ ^'"^ ^'^'"'^'' x^^'^\r^(^. 

Example. — Aii uct-istyit' fror^t, 8 coiunins, 1S4 feel wide, threi- metopes over each inter col urania- 
lion, 21 in all, nud the metope and triglyph 42 and 28, as bel'orr. Then, 

' ^ 21(42 + 2dH-~(6(ri28y = 1^ = '-^^TS o 2 fcct = the d.ameter required. 



SKCTION III.— MOULDINGS, CORNICES, &c. 



MOULDINGS. 



229. — A moulding is so called, because of its being of the 
same determinate shape along its whole length, as though the 
whole of it had been cast in the same mould or form. The regular 
mouldings, as found in remains of ancient architecture, are eight 
in number ; and are known by the following names : 

Annulet, band, cincture, fillet, listel or square. 



Fip. ]23. 



Fig. 124. 



_) Astragal or bead. 



Fig. 125. 



^ Torus or tore. 



L 



Fig: 126. 



Scotia, trochilus or mouth. 



Ovolo, quarter-round or echinas. 

Fig. !27. 



110 



AMERICAN HOUSE-CARPENTER. 




Cavetto, cove or hollow. 



Fij-. 128. 



^ 



Cymatium, or cyma-recta. 



Fig. 129. 



/ 



Inverted cymatiumj or cyma-reversa 



Fit^. 130. 



) 



Some of the terms are derived thus : fillet, from the French 
word Jil, thread. Astragal, from astragalos^ a bone of the heel 
— or the curvature of the heel. Bead, because this moulding, 
when properly carved, resembles a string of beads. Torus, or 
tore, the Greek for rope^ which it resembles, when on the base of 
a column. Scotia, from shotia, darkness, because of the strong 
shadow which its depth produces, and Avhich is increased by the 
projection of the torus above it. Ovolo, from ovum, an egg, 
which this member resembles, when carved, as in the Ionic capi- 
tal. Cavetto, from cavus, hollow. Cymatium, from ^"^/ma^o«, 
a wave. 

230. — Neither of these mouldings is peculiar to any one of the 
orders of architecture, but each one is common to all ; and al- 
though each has its appropriate use, yet it is by no means con- 
fined to any certain position in an assemblage of mouldings. 
The use of the fillet is to bind the parts, as also that of the astra- 
gal and torus, which resemble ropes. The ovolo and cyma-re- 
versa are strong at their upper extremities, and are therefore used 
to support projecting parts above them. The cyma-recta and 
cavetto, being weak at their upper extremities, are not used as 
supporters, but are placed uppermost to cover and shelter the 
other parts. The scotia is introduced in the base of a column, to 



MOULDINGS, CORNICES, &C. 1 I I 

separate the upper and lower torus, and to produce a pleasing 
variety and relief. The form of the bead, and that of the torus, 
is the same ; tlie reasons for giving distinct names to tliem are. 
that the torus, in every order, is always considerably larger than 
the bead, and is placed among the base mouldings, whereas the 
bead is never placed there, but on the capital or entablature ; the 
torus, also, is never carved, whereas the bead is ; and while the 
torus among the Greeks is frequently elliptical in its form, the 
bead retains its circular shape. While the scotia is the reverse of 
the torus, the cavetto is the reverse of the ovolo, and the cyma- 
rectaand cyma-reversa are combinations of the ovolo and cavetto. 

231. — The curves of mouldings, in Roman architecture, were 
most generally composed of parts of circles ; while those of the 
Greeks were almost always elliptical, or of some one of the conic 
sections, bat rarely circular, except in the case of the bead, which 
was always, among both Greeks and Romans, of the form of a 
semi-circle. Sections of the cone afford a greater variety of 
forms than those of the sphere ; and perhaps this is one reason 
why the Grecian architecture so much excels the Roman. The 
quick turnings of the ovolo and cyma-reversa, in particular, when 
exposed to a bright sun, cause those narrow, well-defined streaks 
of light, which give life and splendour to the whole. 

232. — A profile is an assemblage of essential parts and mould- 
ings. That profile produces the happiest effect which is com- 
posed of but few members, varied in form and size, and arranged 
so that the plane and the curved surfaces succeed each other al- 
ternately. 

233. — To describe the Grecian torus and scotia. Join the 
extremities, a and 6, {Fig. 131 ;) and from/, the given projection 
of the moulding, draw/ o, at right angles to the fillets ; from 6, 
draw h A, at right angles to a b ; bisect a b in c ; join / and c, 
and upon c, with the radius, c/ describe the arc, / h, cutting b h 
in h ; through c, draw d e, parallel with the fillets ; make d c and 
c e, each equal to b h ; then d e and a b will be conjugate diame- 



112 



AMERICAN HOUSE-CARPENTER. 




Fig. 131. 



ters of the required ellipse. To describe the curve by intersec- 
tion of lines, proceed as directed at Art. 118 and note; by a 
trammel, see Art. 125 ; and to find the foci, in order to describe it 
with a string, see Art. 115. 




d 


\ 


^ 




a 



Fig. 133 



234. — Fig. 132 to 139 exhibit various modifications of the 
Grecian ovolo, sometimes called echinus. Fig. 132 to 136 eire 



MOULDINGS, CORNICES, &C. 



113 




Fii:. 134. 




Vl'r. 135. 





Fiar.lSe. 



Fig. 13* 




^ b 




Fig. 138. 



Fig. 139. 



elliptical, a b and b c being given tangents to the curve ; parallel 
to which, the semi-conjugate diameters, a d and d c, are drawn. 
In Fig. 132 and 133, the lines, a d and d c, are semi-axes, the 
tangents, a b and b c, being at right angles to each other. To 
draw the curve, see Art. 118. In Fig. 137, the curve is para- 
bolical, and is drawn according to Art. 127. In Fig. 138 and 139, 
the curve is hyperbolical, being described according to Art. 128. 
The length of the transverse axis, a 6, being taken at pleasure 
in order to flatten the curve, a b should be made short in propor- 
tion to a c. 

15 



114 



AMERICAN HOUSE-CARPENTER. 





Fig. 141. 



Fig. 140. 



235. — To describe the Grecian cavetto, {Fig. 140 and 141,) 
having the height and projection given, see Art. 118. 



a 


I 


\iw 


1^ 


Wi 


V 


c 




Fiff. 142. 



Fig. 143. 



236. — To describe the Grecian cyma-recta. When the pro- 
jection is more than the height, as at Fig. 142, make a b equal 
to the height, and divide ah c d into 4 equal parallelograms ; 
then proceed cis directed in note to Art. 118. When the projec- 
tion is less than the height, draw d a, {Fig. 143,) at right angles 
to a b ; complete the rectangle, abed; divide this into 4 equal 
rectangles, and proceed according to Art. 118. 




Fi-. 114 



237. — To describe the Grecian cyma-reversa. When the 



MOULDINGS, CORNICES, &C. 



115 



projection is more than the height, as at Fig. 144, proceed as di- 
rected for the last figure ; the curve being the same as that, the 
position only being changed. When the projection is less than 
the height, draw a d, {Fig. 145,) at right angles to the fillet ; 
make a d equal to the projection of the moulding : then proceed 
as directed for Fig. 142. 

238. — Roman mouldings are composed of parts of circles, and 
have, therefore, less beauty of form than the Grecian. The bead 
and torus are of the form of the semi-circle, and the scotia, also, 
in some instances ; but the latter is often composed of two quaxi- 
rants, having different radii, as at Fig. 1 46 and 147, which re- 
semble the elliptical curve. The ovolo and cavetto are generally 
a quadrant, but often less. When they are less, as at Fig. 150, 
the centre is found thus : join the extremities, a and 6, and bisect 
ahin c ; from c, and at right angles to a b, draw c d^ cutting a 
level line drawn from a in d ; then d will be the centre. This 
moulding projects less than its height. When the projection is 
more than the height, as at Fig. 152, extend the line from c until 





Fig. 146. 



b\s. 14' 



n 




Fif. 148. 



Rf. 14». 



116 



AMERICAN HOUSE-CARPENTER. 





Fig. 151. 



d 


K 




:>^ 




a 



Fig. 152. 




Fig. 153. 







r 


J 











: 



Fig. 154. 



Fig. 155. 





rig. 190. 



Fif. urr 



MOULDINGS, CORNICES, &C. 



117 





Fig. 158. 



Fig. 159. 




fig. 160. 




it cuts a perpendicular drawn from a, as at d; and that will be the 
centre of the curve. In a similar manner, the centres are found 
for the mouldings ait Fig. 147, 151, 153, 156, 157, 158 and 159. 
The centres for the curves at Fig. 160 and 161, are found thus : 
bisect the line, a 6, at c ; upon a, c and b, successively, with a c 
01 c b for radius, describe arcs intersecting at d and d ; then those 
intersections will be the centres. 

239. — Fig. 162 to 169 represent mouldings of modern inven- 
tion. They have been quite extensively and successfully used in 
inside finishing. Fig. 162 is appropriate for a bed-moulding 
under a low, projecting shelf, and is frequently used under man- 
tle-shelves. The tangent, i h, is found thus : bisect the line, a 6, 
at c, and b c aX d; from d, draw d e, at right angles to e b ; from 
b, draw b /, parallel to e d ; upon b, with b d for radius, describe 
the arc, df; divide this arc into 7 equal parts, and set one of the 
parts from 5, the limit of the projection, to o ; make o h equal to 
e; from A, through c, draw the tangent, hi; divide b h, h c,ci 
and i a, each into a like number of equal parts, and draw the in- 



118 



AMERICAN HOUSE-CARPENTER. 




Fig. 162. 




Far 163, 




Fig. 164. 



MOULDINGS, CORNICES, &C. 



119 





Fiff. 165. 



Fig. 166. 





Fig 167. 



Fig. 168, 



Fig. 169 



tersecting lines as directed at Art. 89. If a bolder form is desired, 
draw the tangent, i A, nearer horizontal, and describe an elliptic 
curve as shown in Fig. 131, 164, 175 and 176. Fig. 163 is 
much used on base, or skirting of rooms, and in deep panelling. 
The curve is found in the same manner as that of Fig. 162. In 
this case, however, where the moulding has so little projection 



120 



AMERICAN HOUSE-CARPENTER. 



in comparison with its height, the point, c, being found as in the 
last figure, h s may be made equal to s e, instead of o e as in the 
last figure. Fig. 164. is appropriate for a crown moulding of a 
cornice. In this figure the height and projection are given ; the 
direction of the diameter, a 6, drawn through the middle of 
the diagonal, e /, is taken at pleasure ; and d cis parallel to a 
e. To find the length of d c, draw b h, at right angles to a b ; 
upon 0, with o f for radius, describe the arc,/ h, cutting b h in 
h ; then make o c and o d, each equal to b h* To draw the curve, 
see note to A?^t. 118. Fig. 165 to 169 are peculiarly distinct from 
ancient mouldings, being composed principally of straight lines ; 
the few curves they possess are quite short and quick. 



H. P. 



5 15 



125^1 



2i 11 



9l0i' 



10 



J 



Fig. 170. 



H. P. 



U 15 



Hi 



14i 

13 

iU 



lU 



lO.j 



10 



I 



Fig. 171. 



240.— K^. 170 and 171 are designs for antse caps. The 

* The manner of ascertaining the length of the conjugate diameter, dc/m this figure, 
and also in Fig. 131, 175 and 176, is new, and is important in tliis application. Ic is 
founded upon well-known mathematical principles, viz : All the parallelograms that may- 
be circumscribed about an ellipsis are equal to one another, and consequeudy any one 
is equal to the rectangle of the two axes. And again : the sum of the squares of every 
psdr of conjugate diameters is equal to the sum of the squares of the two axes. 



MOULDINGS, CORNICES, &C. 



121 



diameter of the antse is divided into 20 equal parts, and the height 
and projection of the members, are regulated in accordance with 
those parts, as denoted under ^Tand P, height and projection. 
The projection is measured from the middle of the antas. These 
will be found appropriate for porticos, door- ways, mantle-pieces, 
door and window trimmings, <fec. The height of the antse for 
mantle-pieces, should be from 5 to 6 diameters, having an entab- 
lature of from 2 to 2i diameters. This is a good proportion, it 
being similar to the Doric order. But for a portico these propor- 
tions are much too heavy ; an antae, 15 diameters high, and an en- 
tablature of 3 diameters, will have a better appearance. 

CORNICES. 

241. — Fig. 172, 173 and 174, are designs for eave cornices, 
and Fig. 175 and 176, for stucco cornices for the inside finish of 
rooms. The projection of the uppermost member from the facia, 
is divided into 20 equal parts, and the various members are pro- 
portioned according to those parts, as figured under H and P. 



H. 


P. 














n 


20 


..... , 

1 


5 










J 


f 




i7i 








_J 






J 




3 

8 


16; 












..-.'' ^ 


c^ *'**•. 




1 
25 


1 


) 



















Fig. 172. 

16 



122 



AMERICAN HOUSE-CARPENTER. 



H. P. 

[U;20 



1 
1 



16 



3i 



U 



3i 



u 



^4^ 



U 



2} 



2^ 



Fig. 173. 



:;:^ 




H.P. 



3J20 



3i,16 
1 



. 2 ■ 




Fig. Hi. 



MOULDI.NC.S, CORNICES, &C. 



123 




Fh. 175. 




Fijr. 176. 



124 



AMERICAN HOUSE-CARPENTER. 




b 12 3 4c 

Fig. 177. 



242.-' — To proportioti an eave cornice in accordance with the 
height of the building. Draw the line, a c, {Fig. 177,) and 
make b c and b a, each equal to 18 inches ; from b, draw b d, at 
right angles to a c, and equal in length to | of a c ; bisect b din 
e, and from a, through e, draw a f; upon a, with a c for radius, 
describe the arc, c/, and upon e, with e/for radius, describe the 
'dvcfd; divide the curve, df c, into 7 equal parts, as at 10, 20, 
30, (fee, and from these points of division, draw lines to b c, pa- 
rallel to d b ; then the distance, b 1, is the projection of a cornice 
for a building 10 feet high ; b 2, the projection at 20 feet high ; 
b 3, the projection at 30 feet, ttc. If the projection of a cornice for 
a building 34 feet high, is required, divide the arc between 30 and 
40 into 10 equal parts, and from the fourth point from 30, draw a 
line to the base, b c, parallel with b d ; then the distance of the 
point, at which that line cuts the base, from b, will be the projec- 
tion required. So proceed for a cornice of any height within 70 
feet. The above is based on the supposition that 18 inches is the 
proper projection for a cornice 70 feet high. This, for general 
purposes, will be found correct ; still, the length of the line, b c, 
maybe varied to suit the judgment of those who think differ- 
ently. 

Having obtained the projection of a cornice, divide it into 20 
equal parts, and apportion the several members according to its 
destination — as is shown at Fig. 172, 173 and 174. 



MOULDINGS, CORNICES, &C. 
b 



125 




Fig. 178. 



243. — To proportion a cornice according to a smaller given 
one. Let the cornice at Fig. 178 be the given one. Upon any 
point in the lowest line of the lowest member, as at a, with the 
height of the required cornice for radius, describe an intersecting 
arc across the uppermost line, as at h ; join a and h ; then h 1 will 
be the perpendicular height of the upper fillet for the proposed cor- 
nice, 1 2 the height of the crown moulding — and so of all the 
members requiring to be enlarged to the sizes indicated on this 
line. For the projection of the proposed cornice, draw a d^ at right 
angles to a b, and c d, at right angles to be; parallel with c d, 
draw lines from each projection of the given cornice to the line, 
ad; then e d will be the required projection for the proposed 
cornice, and the perpendicular Imes falling upon e d will indicate 
the proper projection for the members. 

244. — To proportioii a cornice according to a larger given 
one. Let J., {Fig. 179,) be the given cornice. Extend a o to b, 
and draw c d, at right angles to ab; extend the horizontal lines 
of the cornice. A, until they touch o d ; place the height of the 
proposed cornice from o to e, and join / and e ; upon o, with the 
projection of the given cornice, o a, for radius, describe the quad- 
rant, ad; from d, draw d b, parallel iofe; upon o, with o b for 
radius, describe the quadrant, be; then o c will be the proper pro- 
jection for the proposed cornice. Join a and c ; draw lines from the 



126 



AMERICAN HOUSE-CARPENTER. 




Fig. 179. 



projection of the different members of the given cornice to a o, 
parallel to o d ; from these divisions on the line, a o. draw lines 
to the line, o c, parallel to a c ; from the divisions on the line, of, 
draw lines to the line, o e, parallel to the line, f e ; then the di- 
visions on the lines, o e and o c, will indicate the proper height and 
projection for the different members of the proposed cornice. In 
this process, we have assumed the height, o e, of the proposed 
cornice to be given ; but if the projection, o c, alone be given, we 
can obtain the same result by a different process. Thus : upon o, 
with c for radius, describe the quadrant, c b ; upon o, with o a 
for radius, describe the quadrant, a d ; join d and b ; from/, draw 
/ e, parallel to d b ; then o e will be the proper height for the pro- 
posed cornice, and the height and projection of the different mem- 
bers can be obtained by the above directions. By this problem, 
a cornice can be proportioned according to a smaller given one 
as well as to a larger ; but the method described in the previous 
article is much more simple for that purpose. 

245. — To find the angle-bracket for a cornice. Let A, {Fig. 
180,) be the wall of the building, and B the given bracket, which, 
for the present purpose, is turned down horizontally. The angle- 
bracket, C, is obtained thus : through the extremity, a, and paral- 



MOULDINGS, CORNICES, &C. 

A \e 



l^iT 




g Fig. 180 



Fi'j. 181, 



lei with the wal],/c?, drav/ the line, ab ; make e c equal a /, 
and through c, draw c b, parallel with e d ; join d and 6, and from 
the several angular points in B, draw ordinates to cut ^ 6 in 1, 2 
and 3 ; at those points erect lines perpendicular to d b ; from h, 
draw h g, parallel to/ a ; take the ordinates, 1 o, 2 o, &c., at B, 
and transfer them to C, and the angle-bracket, C, will be defined. 
In the same manner, the angle-bracket for an internal cornice, or 
the angle-rib of a coved ceiling, or of groins, as at Fig. 181, can 
be found. 

246. — A level crown moulding being given , to find therakins;' 
m,ouldi7ig and a level return at the top. Let A, {Fig. 182.) be 
the given moulding, and A b the rake of the roof. Divide the 
curve of the given moulding into any number of parts, equal or 
unequal, as at 1, 2, and 3 ; from these points, draw horizontal 
lines to a perpendicular erected from c; at any convenient place 
on the rake, as at 5, draw a c, at right angles to A b ; also, from 
6, draw the horizontal line, b a ; place the thickness, d a, of the 
moulding at A, from b to a, and from a, draw the perpendicular 
line, a e ; from the points, 1, 2, 3, at J., draw lines to C, parallel 
to Ab ; make al, a 2 and a 3, at J5 and at C, equal to a 1, &c., 
at A ; through the points, 1, 2 and 3, at B, trace the curve — this 
will be the proper form for the raking moulding. From 1, 2 and 



128 



AMERICAN HOUSE-CARPENTER. 




Fig 182. 



3. at C. drop perpendiculars to the corresponding ordinates from 
1. 2 and 3, at J. /through the points of intersection, trace the 
curve — this will be the proper tbrm for the return at the top. 



SECTION IV.— FRAMING. 



247. — This subject is, to the carpenter, of the highest impor- 
tance ; and deserves more attention and a larger place in a volume 
of this kind, than is generally allotted to it. Something, indeed, 
has been said upon the geometrical principles, by which the seve- 
ral lines for the joints and the lengths of timber, may be ascer- 
tained ; yet, besides this, there is much to be learned. For how- 
ever precise or workmanlike the joints may be made, what will 
it avail, should the system of framing, from an erroneous position 
of its timbers, <fec., change its form, or become incapable of sus- 
taining even its own weight ? Hence the necessity for a know- 
leds:e of the laws of pressure and the strength of timber. These 
bemg once understood, we can with confidence determine the best 
position and dimensions for the several timbers which compose a 
floor or a roof, a partition or a bridge. As systems of framing 
are more or less exposed to heavy weights and strains, and, in 
case of failure, cause not only a loss of labour and material, but 
frequently that of life itself, it is very important that the materials 
employed be of the proper quantity and quality to serve their des- 
tination. And, on the other hand, any superfluous material is not 
only useless, but a positive injury, it being an unnecessary load 
upon the points of support. It is necessary, therefore, to know 

17 



130 



AMERICAN HOUSE-CARPENTER. 



the least quantity of timber that will suffice for strength. The 
greatest fciult in framing is that of using an excess of material. 
Economy, at least, would seem to require that this evil be abated. 

Before proceeding to consider the principles upon which a sys- 
tem of framing should bo constructed, let us attend to a few of 
the elementary laws iji Mechanics ^ w^hich will be found to be of 
great value in determining those principles. 

248. — Laws of Pressure. (1.) A heavy body always 
exerts a pressure equal to its own weiglit in a vertical direction. 
Example: Suppose an iron ball, weighing lUU lbs., be supported 
upon the top of a perpendicular post, {Fig. 196;) then the 
pressure exerted upon that post will be equal to the weight of the 
ball; viz., 100 lbs. (2.) But if two inclined posts, [Fig. 183,) 
be substituted for the perpendicular support, the united pressures 
upon these posts will be more than equal to the Vv^eight, and will 
be in proportion to their position. The farther apai t their feet are 
spread the greater will be the pressure, nnd vice versa. Hence 
tremendous strains m.ay be exerted by a comparatively small 
weight. And it follows, therefore, that a piece of timber intend- 
ed for a strut or post, should be so placed that its axis may coin- 
cide, as near as possible, with the direction of the pressure. The 
direction of the pressure of the weight, W, {Fig. 183,) is in the 
vertical line, h d ; and the weight, W^ would fall in that line, it 
the two posts were removed, hence the best position for a support 



w 




Fig. 183. 



FRAMING. 



131 



for the weight would be in that line. But, as it rarely occurs in 
systems of framing that weights can be supported by any single 
resistance, they requiring generally two or more supports, (as in 
the case of a roof supported by its rafters,) it becomes important, 
therefore, to know the exact amount of pressure any certain 
weight is capable of exerting upon oblique supports. This can 
be ascertained by the following process. 

Let a h and h c, {Fig. 183,) represent the axes of two sticks of 
timber supporting the weight, W ; and let the weight, W, be 
equal to 6 tons. Make the vertical line, h d, equal to 6 inches ; 
from d, draw df, parallel to a b, and d e, parallel Xo c b ; then 
the line, b e, will be found to be Sg inches long, which is equal tc 
the number of tons that the weight, W, exerts upon the post, a b. 
The pressure upon the other post is represented by bf, which in 
this case is of the same length as b e. The posts being inclined 
at equal angles to the vertical line, b d, the pressure upon them is 
equal. Thus it will be found that the weight, which weighs 
only 6 tons, exerts a pressure of 7 tons ; the amount being in- 
creased because of the oblique position of the supports. The 
lines, e b, bf,fd and d e, compose what is called the parallelo- 
gram of forces. The oblique strains exerted by any one force, 
therefore, may always be ascertained, by making b coequal, (upon 
any scale of equal parts,) to the number of lbs., cwts.. or tons 
contained in the weight, W^ and b e will then represent the num- 
ber of lbs., cwts., or tons with which the timber, a 6, is pressed, 
and bf that exerted upon b c. 




Fig. 184 



132 



AMERICAN HOUSE-CARPENTER. 



Correct ideas of the comparative pressure exerted upon timbers 
according to their position, will be readily formed by drawing 
various designs of framing, and estimating the several strains in 
accordance with these principles. In Fig. 184, the struts are 
framed into a third piece, and the weight suspended from that. 
The struts are placed at a different angle to show the diverse 
pressures. The length of the timber used as struts, does not 
alter the amount of the pressure. But it may be observed that 
long timbers are not so capable of resistance as short ones. 




Fig. 165. 



249. — In Fig. 185, the weight, W^ exerts a pressure on the 
struts in the direction of their length ; their feet, w, w, have, there- 
fore, a tendency to move in the direction, n o, and would so move, 
were they not opposed by a sufficient resistance from the blocks, 
A and A. If a piece of each block be cut off at the horizontal 
line, a ri, the feet of the struts would slide away from each other 
along that line, in the direction, n a ; but if, instead of these, two 
pieces were cut off at the vertical line, n 6, then the struts would 
descend vertically. To estimate the horizontal and the vertical 
pressures exerted by the struts, let w o be made equal (upon any 
scale of equal parts) to the number of tons (or pounds) with 
.vhich the strut is pressed ; construct the parallelogram of forces 



FRAMING. 



133 



by drawing© e parallel to a n, and 0/ parallel to 6 ii ; then 71 f, 
(by the same scale,) shows the number of tons (or pounds) pres- 
sure that is exerted by the strut in the direction, n a, and n e 
shows the amount exerted in the direction, ?i h. By constructing 
designs similar to this, giving various and dissimilar positions to 
the struts, and then estimating the pressures, it will be found in 
every case that the horizontal pressure of one strut is exactly 
equal to that of the other, however much one strut may be in- 
clined more than the other ; and also, that the united vertical 
pressure of the two struts is exactly equal to the weight, W. (In 
this calculation, the weight of the timbers is not taken into con- 
sideration.) 

250. — Suppose that the two struts, B and jB, {Fig. 185,) were 
rafters of a roof, and that instead of the blocks, A and A^ the walls 
of a building were the supports : then, to prevent the walls from 
being thrown over by the thrust of B and B^ it would be desira- 
ble to remove the horizontal pressure. This may be done by uni- 
ting the feet of the rafters with a rope, iron rod, or piece of tim- 
ber, as in Fig. 186. This figure is similar to the truss of a roof. 




Fig. 186. 



The horizontal strains on the tie-beam, tending to pull it asunde' 
in the direction of its length, may be measured at the foot of th^ 



134 



AMERICAN HOUSE-CARPENTER. 



rafter, as was shown at Fig. 185 ; but it can be more readily 
and as accurately measured, by drawing from /and e horizontal 
lines to the vertical line, h d, meeting it in o and o; then/ o will be 
the horizontal thrust at B, and e ooX A ; these will be found to 
equal one another. When the rafters of a roof are thus connected, 
all tendency to thrust the walls horizontally is removed, the only 
pressure on them is in a vertical direction, being equal to the 
weight of the roof and whatever it has to support. This pres- 
sure is beneficial rather than otherwise, as a roof thus formed 
tends to steady the walls. 




Fig. 188. 



251 — Fig. 187 and 188 exhibit methods of framing for sup- 
porting the equal weights, Tl^^and W. Suppose it be required to 
measure and compare the strains produced on the pieces, A B 
and A C. Construct the parallelogram of forces, e h f d^ ac- 
cording to Art. 248. Then h f will sliow the strain on A B, and b 
5 the strain on A C. By comparing the figures, b d being equal 
in each, it will be seen that the strains in Fig. 187 are about three 



FRAMING. 135 

times as great as those in Fig. 188 : the position of the pieces, 
A B and A C, in Fig. 188, is therefore far preferable. 

This and the preceding examples exemplify, in a measure, the 
resolution of forces ; viz., the finding of two or more forces, which, 
acting in different directions, shall exactly balance the pressure 
of any given sijigle force. Thus, in Fig. 185, supposing the 
weight, Wj to be the greatest force that the two timbers, in their 
present position, are capable of sustaining, then the weight, W, 
is the given force, and the timbers are the two forces just equal to 
the given force. 




C Fig. 189 

252. — The composition of forces consists in ascertaining the 
direction and amount of one force, which shall be just capable of 
balancing two or inore given forces, acting in different directions. 
This is only the reverse of the resolution of forces, and the two 
are founded on one and the same principle, and may be solved in 
the same manner. For example ; let A and B, {Fig. 189,) be 
two pieces of timber, pressed in the direction of their length to- 
wards b — A by a force equal to 6 tons weight, and B equal to 9. 
To find the direction and amount of pressure they would unitedly 
exert, draw the lines, b e and b f in a line with the axes of the 
timbers, and make b e equal to the pressure exerted by B, viz., 9 ; 
also make b f equal to the pressure on A, viz., 6, and complete 
the parallelogram of forces, ebfd; then b d, the diagonal of th^ 



136 



AMERICAN HOUSE-CARPENTER. 



parallelogram, will be the direction, and its length will be the 
amoujitj of the united pressures of A and of B. The line, b d, is 
termed the resultant of the two forces, h f and he. HA and B are to 
be supported by one post, C, the best position for that post will be 
in the direction of the diagonal, h d ; and it will require to be 
sufficiently strong to support the united pressures of A and of B 




't*^9 



Fig. 190. 



253. — Another example : let Fig. 190 represent a piece oi 
framing commonly called a crane, which is used for hoisting 
heavy weights by means of the rope, Bh f, which passes over a 
pulley at h. This is similar to Fig. 187 and 188, yet it is mate- 
rially different. In those figures, the strain is in one direction 
only, viz., from b to d ; but in this there are two strains, from A 
to B and from A to W. The strain in the direction, A B, is evi- 
dently equal to that in the direction, A W. To ascertain the best 
position for the strut, A C, make b e equal to b f, and complete 
the parallelogram of forces, e bfd; then draw the diagonal, b d, 
and it will be the position required. Should the foot, C, of the 
strut be placed either higher or lower, the strain on A C would be 
increased. In constructing cranes, it is advisable, in order that 
the piece, B A, may be under a gentle pressure, to place the foot 
of the strut a trifle lower than where the diagonal, 6 d, would in- 
dicate^ but never higher 



FRAMING. 



137 



r\ 



•w/lltv^ 






B 


L 








flc 








^^ . 




Sa«», 




v_y 


'^V/V 



w 



254. — Ties and Struts. Timbers in a state of tension are 
called ties, while such as are in a state of compression are termed 
struts. This subject can be illustrated in the following manner. 

Let A and B, [Fig. 191,) represent beams of timber supporting 
the weights, TF, W and W; A having but one support, which is 
in the middle of its length, and B two, one at each end. To 
show the nature of the strains, let each beam be sawed in the 
middle from a to b. The effects are obvious : the cut in the 
beam, A, will open, whereas that in B will close. If the weights 
are heavy enough, the beam, A, will break at b ; while the cut in 
B will be closed perfectly tight at a, and the beam be very little 
injured by it. But if, on the other hand, the cuts be made in the 
bottom edge of the timbers, from c tob, B will be seriously in- 
jured, while A will scarcely be affected. By this it appears evident 
that, in a piece of timber subject to a pressure across the direction 
of its length, the fibres are exposed to contrary strains. If the tim- 
ber is supported at both ends, as at B, those from the top edge down 
to the middle are compressed in the direction of their length, while 
those from the middle to the bottom edge are in a state of tension ; 
but if the beam is supported as at J., the contrary effect is produced ; 
while the fibres at the middle of either beam are not at all strained. 
The strains in a framed truss are of the same nature as those in 
a single beam. The truss for a roof, being supported at each end, 
has its tie-beam in a state of tension, while its rafters are com- 
pressed in the direction of their length. By this, it appears highly 
important that pieces in a state of tension should be distinguished 

18 



13S AMERICAN HOUSE-CARPENTER. 

from siich as are compressed, in order that the former may be pre- 
served continuous. A strut may be constructed of two or more 
pieces; yet, where there are many joints, it will not resist com- 
ression so firmly. 

255. — To distinguish ties from struts. This may be done 
by the following rule. In Fig. 183, the timbers, a h and h c, are the 
sustaining forces, and the weight, W, is the straining force ; and, 
if the support be removed, the straining force would move from 
the point of support, 6, towards d. Let it be required to ascer- 
tain whether the sustaining forces are stretched or j^ressed by the 
straining force. Rule : upon the direction of the straining force, 
b c?, as a diagonal, construct a parallelogram, e b f d, whose sides 
shall be parallel with the direction of the sustaining forces, a b 
and c 6 ; through the point, 5, draw a line, parallel to the diag- 
onal, cf; this may then be called the dividing line between ties 
and struts. Because all those supports which are on that side of 
the dividing line, which the straining force would occupy if unre- 
sisted, are compressed, while those on the other side of the divi- 
ding line are stretched. 

In Fig. 183, the supports are both compressed, being on that 
side of the dividing line which the straining force would occupy 
if unresisted. In Fig. 187 and 188, in which A B and A C 
are the sustaining forces, A C is compressed, whereas A B is in 
a state of tension ; A C being on that side of the line, h i, which 
the straining force would occupy if unresisted, and A B on the 
opposite side. The place of the latter might be supplied by a 
chain or rope. In Fig. 186, the foot of the rafter at A is sus- 
tained by two forces, the wall and the tie-beam, one perpendicular 
and the other horizontal : the direction of the straining force is 
indicated by the line, b a. The dividing line, h i, ascertained 
by the rule, shows that the wall is pressed and the tie-beam 
stretched. 

256. — Another example : let E A B P, {Fig. 192,) represent 
a gate, supported by hinges at A and E. In this case, the strain- 




Fi?. 19-2. 



i?ig force is the weight of the materials, and the direction ol 
course verticaL Ascertain the dividing line at the several points, 
G, B, I, J, Hnud F. It will then appear that the force at G is 
sustained by A G and G E, and the dividing line shows that the 
former is stretched and the latter compressed. The force at H is 
supported by A Hand HE — the former stretched and the latter 
compressed. The force at B is opposed by H B and A B, one 
pressed — the other stretched. The force at jPis sustained by G 
Fwiid F E, G Fhe'mg stretched and FE pressed. By this it 
appeals that A B is in a state of tension, and E F, of compres- 
sion; also, that AHnud G Fare stretched, wiiile B H and G 
E are compressed : which sliows tlie necessity of having A H 
and G F, each in oue whole length, while B Hand G E may 
be, as they are shown, each in two pieces. The force at /is sus- 
tained by G J and J H, the former stretched and the latter com- 
pressed. The piece, C D, is neither stretched nor pressed, and 
could be dispensed with if the joinings at J and i could be made 
as effectually without it. In case A B should fail, then C D 
would be in a state of tension. 

257. — The pressure of inclined beams. The centre of gravi- 
ty of a uniform prism or cylinder, is in its axis, at the middle of 
its length. In irregular bodies with plain sides, the cent'e ol 



140 



AMERICAN HOUSE-CARPENTER. 



gravity may be found by balancing them upon the edge of a prism 
in two positions, making a line each time upon the body in a line- 
with the edge of the prism, and the intersection of those lines 
will indicate the point required. 




Fig. 193. 



An inclined post or strut, supporting some heavy pressure ap- 
plied at its upper end, as at Fig. 186, exerts a pressure at its foot 
in the direction of its length, or nearly so. But when such a 
beam is loaded uniformly over its whole length, as the rafter of a 
roof, the pressure at its foot varies considerably from the direction 
of its length. For example, let A B, {Fig. 193,) be a beam lean- 
ing against the wall, B c, and supported at its foot by the abut- 
ment, A, in the beam, A c, and let o be the centre of gravity of the 
beam. Through o, draw the vertical line, b d, and from B, draw 
the horizontal Ime, B b, cattirig b d in b ; join b and A^ and b A 
will be the direction of the thrust. To prevent the beam from 
loosing its footing, the joint at A should be made at right angles 
to b A. The amount of pressure will be found thus : let b d, 
(by any scale of equal parts,) equal the number of tons, cwts., 
or pounds weight upon the beam, A B ; draw d e, parallel to B 
b ; then b e, (by the same scale,) equals the pressure in the direc- 
tion, b A ; and e d, the pressure against the wall at B — and also 
the horizontal thrust at A^ as these are always equal in a construc- 
tion of this kind. Fig. 194 represents two equal beams, sup- 
ported at their feet by the abutments in the tie-beam. This case 
is similar to the last ; for it is obvious that each beam is in pre- 
cisely the position of the beam in Fig. 193. The horizontal 



FRAMING. 



141 




Fig. 194. 



pressures at 5, being equal and opposite, balance one another; 
and their horizontal thrusts at the tie-beam are also equal. (See 
Art. 250 — Fig. 186.) When the inclination of a roof, [Fig. 
194,) is one-fourth of the span, or of ashed, [Fig. 193,) is one-half 
the span, the horizontal thrust of a rafter, whose centre of gravity- 
is at the middle of its length, is exactly equal to the weight dis- 
tributed uniformly over its surf^ice. The inclination, in a rafter 
uniformly loaded, which will produce the least oblique pressure, 
{6 e. Fig. 193,) is 35 degrees and 16 minutes. 




^■^rA 



Fig. 195. 



%8. — In shed, or lean-to roofs, as Fig. 193, the horizontal 
j>:it*<5sure will be entirely removed, if the bearings of the rafters, as 
A, Bj {Fig. 195,) are made horizontal — provided, however, that 
the rafters and other framing do not bend between the points of 
support. If a beam or rafter have a natural curve, the convex 
or rounding edge should be laid uppermost. 

259. — A beam laid horizoptaPy supported at each end and 
uniformly loaded, is subjec* to tJi«' greatest strain at the middle 



142 



AMERICAN HOUSE-CARPENTER. 



of its length. The amount of pressure at that point is equal to 
half of the whole load sustained. The greatest strain coming 
upon the middle of such a beam, mortices, large knots and othei 
defects, should be kept as far as possible from that point ; and, in 
resting a load upon a beam, as a partition upon a floor beam, the 
weight should be so adjusted that it will bear at or near the ends. 
(See Art. 282.) 

260. — The resistance of tvtnher. When the stress that a 
given load exerts in any particular direction, has been ascertain- 
ed, before the proper size of the timber can be determined for the 
resistance of that pressure, the strength of the kind of timber to 
be used must be known. The following rules for calculating the 
resistance of timber, are based upon the supposition that the tim- 
ber used be of what is called " merchantable" quality — that is. 
strait-grained, seasoned, and free from large knots, splits, decay, 
&c. 





Fig. 196. 



Fig. 197 




Fig. 198. 



The strength of a })iece of timber, is to be considered in ac* 
cordance witti the direction in which the strain is applied upon 



FRAMING. 143 

it. When it is compressed in the direction of its length, as in 
Fig. 196j its strength is termed the resistance to compression. 
When the force tends to pull it asunder in the direction of its 
length, (^, Fig. 197",) it is termed the resistance to tension. 
And when strained by a force tending to break it crosswise, as at 
Fig. 198, its strength is called the resistance to cross strains. 

261. — Resistance to compression. When the height of a 
piece of timber exceeds about 10 times its diameter if round, or 
10 times its thickness if rectangular, it will bend before crushing. 
The first of the following cases, therefore, refers to such posts as 
would be crushed if overloaded, and the other two to such as 
would hend before crushing. In estimating the strength of tim- 
ber for this kind of resistance, it is provided in the following 
rules that the pressure be exactly in a line v^^ith the axis of the 
post. 

Case 1. — To find the area of a post that will safely bear a given 
weight — when the height of the post is less than 10 times its least 
thickness. Ride. — Divide the given weight in pounds by 1000 
for pine and 1400 for oak, and the quotient will be the least area 
of the post in inches. This rule requires that the area of the 
abutting surface be equal to the result : should there be, there- 
fore, a tenon on the end of the post, this quotient will be too small. 
Example. — What should be the least area of a pine post that will 
safely sustain 48,000 pounds ? 48,000, divided by 1000, gives 
48 — the required area in inches. Such a post may be 6x8 
inches, and will bear to be of any length within 10 times 6 inches, 
its least thickness. 

Case 2. — To find the area of a rectangular post that will 
safely bear a given weight — when its height is 10 times its least 
thickness or more. Rule. — Multiply the given weight or pres- 
sure in pounds by the square of the length in feet ; and multi- 
ply this product by the decimal, '0015, for oak, '0021, for pitch 
pine and '0016 for white pine ; then divide this product by the 
breadth in inches, and the cube-root of the quotient will be the 



144 AMERICAN HOUSE-CARPENTER. 

thickness in inches. Example. — What should be the thickness 
of a pine post, 8 feet high and 8 inches wide, in order to support 
a weight of 12 tons, or 26,880 pounds ? The square of the length 
is 64 feet; this, multiplied by the weight in pounds, gives 
1,730,320; this product, multiplied by the decimal, '0016, gives 
2768-512 ; and this again, divided by the breadth in inches, gives 
346-064 ; by reference to the table of cube-roots in the appendix, 
the cube-root of this number will be found to be 7 inches large^ — 
which is the thickness required. The stiffest rectangular post is 
that in which the sides are as 10 to 6. 

Case 3.— To find the area of a row?ic?, or cylindrical, post, that 
will safely hear a given weight — when its height is 10 times its 
least diameter or more. Rule. — Multiply the given weight or 
pressure in pounds by 1*7, and the product by -0015 for oak, -0021 
for j)itch pine and '0016 for white pine ; then multiply the square- 
root of this product by the height in feet, and the square-root of 
the last product will be the diameter required, in inches. Exam- 
ple. — What should be the diameter of a cylindrical oak post, 8 
feet high, in order to support a weight of 12 tons, or 26,880 
pounds ? This weight in pounds, multiplied by 1*7, gives 45,696 ; 
and this, by '0015, gives 68-544 ; the square-root of this product 
is (by the table in the appendix) 8*28, nearly — which, multiplied 
by 8, gives 66*24 ; the square-root of this number is 8*14, nearly ; 
therefore, 8-14 inches is the diameter required. 

Experiments have shown that the pressure should never be 
more than 1000 pounds per square inch on a joint in yellow pine 
— when the end of the grain of one piece is pressed against the 
side of the grain of the other. 

262. — Resistance to tension. A bar of oak of an inch square, 
pulled in the direction of its length, has been torn asunder by a 
weight of ... - 11,500 lbs. 

Of white pine - - - 11,000 

Of pitch pine - - - 10,000 



FRAMING. 145 

Therefore, when the strain is applied in a line with the axis of 
the piece, the following rule must be observed. 

To find the area of a piece of timber to resist a given strain in 
the direction of its length. Rule. — Divide the given weight to 
be sustained, by the weight that will tear asunder a bar an inch 
square of the same kind of wood, (as above.) and the product will 
be the area in inches of a piece that will just sustain the given 
weight ; but the area should be at least 4 times this, to safely 
sustain a constant load of the given weight. Example. — What 
should be the area of a stick of pitch pine timber, which is re- 
quired to sustain safely a constant load of 60,000 pounds ? 60,000, 
divided by 10,000, (as above,) gives 6, and this, multiplied by 4, 
give 24 inches— the answer. 

263. — Resistance to cross strains. To find the scantling of a 
piece of timber to sustain a given weight, when such piece is 
supported at the ends in a horizontal position. 

Case 1. — When the breadth is given. Rule. — Multiply the 
square of the length in feet by the weight in pounds, and this 
product by the decimal, '009, for oak, 'Oil for white pine and '016 
for pitch pine ; divide the product by the breadth in inches, and 
the cube-root of the quotient will be the depth required in inches. 
Example. — What should be the depth of a beam of white pine^ 
having a bearing of 24 feet and a breadth of 6 inches, in order to 
support 900 pounds ? The square of 24 is 576, and this, multiplied 
by 900, gives 518-400; and this again, by -Oil, gives 5702-400 ; 
this, divided by 6, gives 950*400 ; the cube-root of which is 9*83 
inches — the depth required. 

Case 2. — When the depth is given. Rule. — Multiply the 
square of the length in feet by the weight in pounds, and multi- 
ply this product by the decimal, -009, for oak, -Oil for white pine 
and '016 for pitch pine ; divide the last product by the cube of 
the depth in inches, and the quotient will be the breadth in inches 
required. Exarnple. — What should be the breadth of a beam of 
oak, having a bearing of 1 6 feet and a depth of 12 inches, in 

19 



146 AMERICAN HOUSE-CARPENTER. 

order to support a weight of 4000 pounds? The square of 16 is 
256, which, multiplied by 4000, gives 1,024,000 ; this, multiplied 
by '009, gives 9216 ; and this again, divided by 1728, the cube of 
12, gives 5-g inches — which is the breadth required. 

Case 3. — When the breadth bears a certain proportion to the 
depth. When neither the breadth nor depth is given, it will be 
best to fix on some proportion which the breadth should have to 
the depth ; for instance, suppose it be convenient to make the 
breadth to the depth as 0*6 is to 1, then the rule would become as 
follows : Rule. — Multiply the weight in pounds by the decimal, 
•009, for oak, 'Oil for white pine and '016 for pitch pine; divide 
the product by 0*6, and extract the square-root ; multiply this root 
by the length in feet, and extract the square-root a second time, 
v/hich will be the depth in inches required. The breadth is 
equal to the depth multiplied by the decimal, 0*6. It is obvious 
that any other proportion of tiie breadth and depth may be ob- 
tained by merely changing the decimal, 0'6, in the rule. Exam- 
ple. — What should be the depth and breadth of a beam of pitch 
pine, having a proportion to one another as 06 to 1, and a bearing 
of 22 feet, in order to sustain a ton weight, or 2240 pounds !■ 
This, multiplied by '016, gives 35*84, which, divided by 0-G, 
gives 59'73 ; the square-root of this is 7'7, which, multiplied by 
22, the length, gives 169*4 ; the square-root of this is 13 — which 
is the depth required. Then 13, multiplied by 0'6, gives 7*8 
inches — the required breadth. 

Case 4. — When the beam is inclined, as A B, Fig. 193. 
Rule. — Multiply together the weight in pounds, the length of the 
beam in feet, the horizontal distance, A c. between the supports, 
in feet, and the decimal, '0095 for oak, -Oil for white pine, and 
•016 for pitch pine ; divide this product by 0*6, and the fourth 
root of the quotient will give the depth in inches. The breadth 
is equal to the depth multiplied by the decimal, 0*6. Example. — 
What should be the size of an oak beam, the sides to bear a pro- 
portion to ci.e another as 0*6 to 1, in order to support a ton weight 



FRAMING. 



147 



or 2240 pounds, the beam being inclined so that, its length being 
20 feet, its horizontal distance between the points of support will 
be 16 feet? 2240, multiplied by 20, gives 44,800, which, multi- 
plied by 16, gives 716,800 ; and this again, by the decimal, -009, 
gives 6451-2 ; this last, divided by 0-6, gives 10,752, the fourth 
root of which is 1018, nearly ; and this, multiplied by 0*6, gives 
6*1 ; therefore, the size of the beam should be 10*18 inches by 
6-1 inches. 




Fig. 199. 



264. — To ascertain the scantling of the stiffest beam that 
can he cut from a cj/linder. Let d a c b, [Fig- 199,) be the sec- 
tion, and e the centre, of a given cylinder. Draw the diameter, 
a b ; upon a and 6, with the radius of the section, describe the 
arcs, d e and e c ; join d and «, a and c, c and h. and h and d ; 
then the rectangle, d a cb^ will be a section of the beam required. 

265. — The greater the depth of a beam in proportion to the 
thickness, the greater the strength. But when the difference be- 
tween the depth and the breadth is great, the beam must be 
stayed, (as at Fig. 202,) to prevent its falling over and breaking 
sideways. Their shrinking is another objection to deep beams ; 
but where these evils can be remedied, the advantage of increas- 
ing the depth is considerable. The following rule is, to find the 
strongest form for a beam out of a given quantity of timber. 
Rule. — Multiply the length in feet by the decimal, 0*6, and divide 
the given area in inches by the product ; and the square of the 
quotient will give the depth in inches. Example. — What is the 
strongest form for a beam whose given area of section is 48 



148 AMERICAN HOUSE-CARPENTER. 

ii'icheSj and length of bearing 20 feet ? The length m feet, 20. 
multiplied by the decimal, 0*6, gives 12; the given area in inches, 
48, divided by 12, gives a quotient of 4, the square of which is 
16 — this is the depth in inches ; and the breadth must be c 
inches. A beam 1(3 inches by 3 would bear twice as much as a 
square beam of the same area of section; which shows how im- 
portant it is to make beams deep and thin. In many old build- 
in[;s, and even in new ones, in country places, the very reverse of 
this has been practised ; the principal beams being oftener laid 
on the broad side than on the narrower one. 

266. — Systems of Framing. In the various parts of framing 
known as floors, partitions, roofs, bridges, &c., each has a specific 
object ; and, in all designs for such constructions, this object 
should be kept clearly in view ; the various parts being so dis- 
posed as to serve the design with the least quantity of material. 
The simplest form is the best, not only because it is the most 
economical, but for many other reasons. The great number of 
joints, in a complex design, render the construction liable to de- 
rangement by multiplied compressions, shrinkage, and, in conse- 
quence, highly increased oblique strains ; by which its stability 
and durability are greatly lessened. 

FLOORS. 

267. — Floors have been constructed in various ways, and are 
known as smgle-joisted, double., and framed. In a single- 
joisted floor, the timbers, or fioor-joists, are disposed as is shown in 
Fig. 200. Where strength is the principal object, this manner 
of disposing the fioor-joists is far preferable ; as experiments have 
proved that, with the same quantity of material, single-joisted 
floors are much stronger than either double or framed floors. 
To obtain the greatest strength, the joists should be thin and 
deep. 

268. — To find the depth of a joist, the length of hearing 
and thickness being given, when the distance from centres is 



FRAMING. 



149 




Fig. 20a 



12 inches. Rule. — Divide the square of the length in feet, by 
the breadth in inches ; and the cube-root of the quotient, multi- 
pUed by 2-2 for pine, or 2*3 for oak, will give the depth in inches. 
Example. — What should be the depth of floor-joists, having a 
bearing of 12 feet and a thickness of 3 inches, when said joists 
are of pine and placed 12 inches from centres ? The square of 
12 is 144, which, divided by 3, gives 48 ; the cube-root of this 
number is 3*63, which, multiplied by 2-2, gives 7*986 inches, 
the depth required ; or 8 inches will be found near enough for 
practice. 

269. — Where chimneys, flues, stairs, <fec., occur to interrupt 
the bearing, the joists are framed into a piece, (6, Fig. 201,) 
called a trimmer. The beams, a, a, into which the trimmer is 
framed, are called tri?n7?iing-b earns, trimming-joists, or car- 
riage-beams. They need to be stronger than the common joists, 
in proportion to the number of beams, c, c, which they support. 
The trimmers have to be made strong enough to support half the 
weight which the joists, c, c, support, (the wall, or another trim- 
mer, at the other end supporting the other half,) and the carriage- 



160 



AMERICAN HOUSE-CARPENTER. 




beams must each be strong enough to support half the weight 
which the trimmer supports. In calculating for the dimensions 
of floor-timbers, regard must be had to the fact that the weight 
which they generally support — such as persons of 150 pounds 
moving over the floor — exerts a much greater influence than 
equal weights at rest. When the trimmer, b, is not more dis- 
tant from the bearing, d, than is necessary for ordinary hearths, 
&c., it will be sufficient to add I of an inch to the thickness of 
the carriage-beam for every joist, c, that is supported. Thus, if 
the thickness of c is 3 inches, and the number of joists supported 
be 6, add 6 eighths, or f of an inch, making the carriage-beams 
3| inches thick. It is generally the practice in dwellings to make 
the carriage-beam, in all situations, one inch thicker than the 
common joists. But it is well to have a rule for determining the 
size more acciuately in extreme cases. 

270. — When the bearing exceeds 8 feet, there should be struts, 
as a and a, [Fig- 202,) well nailed between the joists. These 
will prevent the turning or twisting of the floor-joists, and will 
greatly stifien the floor. For, in the event of a heavy weight 
resting upon one of the joists, these struts will prevent that joist 
from settling below the others, to the injury of the plastering 



FRAMING. 



16' 




Fig. 202. 



upon the undex^'ide. When the length of bearing is great, struts 
should be inserted at about every 4 feet. 

271. — Single-joisted floors may be constructed for as great a 
length of bearing as timber of sufficient depth can be obtained ; 
but, in such cases, where perfect ceilings are desirable, either 
double or framed floors are considered necessary. Yet the ceil- 
ings under a single-joisted floor may be rendered more durable by 
cross-furring^ as it is termed — which consists of nailing a series 
of narrow strips of board on the under edge of the beams and at 
right angles to them. To these, instead of the beams, the laths 
are nailed. The strips should be not over 2 inches wide — enough 
to join the laths upon is all that is wanted in width — and not 
more than 12 inches apart. It is necessary that all furring for 
plastering be narrow, in order that the mortar may have a sufli- 
cient clinch. 

When it is desirable to prevent the passage of sound, the open- 
ings between the beams, at about 3 inches from the upper edge, 
are closed by short pieces of boards, which rest on cleets nailed 
to the beam along its whole length. This forms a floor upon 
which mortar is laid to the depth of about 2 inches, leaving but 
about half an inch from its upper surface to the under side of the 
floor-plank. 

272. — Double floors. A double floor consists, as at Fig. 203, 
of three tiers of joists or timbers ; viz., bridging-joists, a, a, 
binding-joists, 6, 6, and ceiling-joists, c, c. The binding-joists 



152 



AMERICAN HOUSE-CARPENTEB. 




Fig. 203. 



are the principal support, and of course reach from wall to wall. 
The bridging-joists, which support the floor-plank, are laid upon 
the binding-joists, to which they are nailed; sometimes they are 
notched into the binding-joists, but they are sufliciently firm 
when well nailed. The ceiling-joists are notched into the under 
side of the binders, and nailed ; they are the support of the lath 
and plastering. 

273. — Binders are laid 6 feet apart. At this distauce the fol- 
lowing rules will give the scantling. 

Case 1.— To find the depth of a binding-joist, the length and 
breadth being given. jR/^/e.— Divide the square of the length in 
feet, by the breadth in inches ; and the cube-root of the quotient, 
multiplied by 3-42 for pine, or by 3'53 for oak, will give the depth 
in inches. Example.— What should be the depth of a binding- 
joist, having a length of 12 feet and a breadth of 6 inches, when 
the kind of timber is pine ? The square of 12 is 144, which, di- 
vided by 6, gives 24 ; the cube-root of this is 2-88, which, multi- 
plied by 3-42, gives 9*85, the depth in inches. 

Case 2.— To find the breadth, when the depth and length are 
§iven. Rule.—'DiYide the square of the length in feet, by the 



FRAMING. 153 

cube of the depth in inches ; and multiply the quotient by 40 for 
pine, or by 44 for oak, which will give the breadth in inches. 
Example. — What should be the breadth of a binding-joist, hav- 
ing a length of 12 feet and a depth of 10 inches, when the kind 
of wood is pine ? The cube of 10 is 1000 ; the square of 12 is 
144 ; this, divided by 1000, gives a quotient of -144 ; and this 
quotient, multiplied by 40, gives 5*76, the breadth in inches. 

274. — Bridging-joists are laid from 12 to 20 inches apart. The 
scantling may be found by the rule at Art. 268. 

275. — Ceiling-joists are generally placed 12 inches apart from 
centres. They are arranged to suit the length of the lath ; this 
being, in most cases, 4 feet long. What is said at Art. 271, in 
regard to the width of furring for plastering, will apply to the 
thickness of ceiling-joists. 

To find the depth of a ceiling-joist, when the length of bearing 
and thickness are given. Rule. — Divide the length in feet by 
the cube-root of the breadth in inches ; and multiply the quotient 
by 0'64 for pine, or by 0"67 for oak, which will give the depth in 
inches. Examiile. — What should be the depth of a ceiling-joibt 
of pine, when the length of bearing is 6 feet and the thickness 2 
inches ? The length in feet, 6, divided by the cube-root of the 
breadth in inches, 1*26, gives a quotient of 4*76, which, being 
multiplied by the decimal, 0*64, gives 3 inches, the depth re- 
quired. 

When the thickness of a ceiling-joist is 2 inches, the depth in 
inches will be equal to half the length of bearing in feet. Thus, 
if the bearing is 6 feet, the depth will be 3 inches ; bearing 8 
feet, depth 4 inches, (fcc. 

276. — Framed floors. When a good ceiling is required, and 
the distance of bearing is great, the binding-joists, instead of 
reaching from wall to wall, are framed into girders. These are 
heavy timbers, as c?, {Fig. 204,) which reach from wall to wall, 
being the chief support of the floor. Such an arrangement is 
termed di framed floor. The binding, the bridging and the ceil 

20 



154 



AMERICAN HOUSE-CARPENTER. 




Fig. 204. 



ing-joists in these, are the same as those in double floors just 
described. The distinctive feature of this kind of floor is the 
girder. 

277. — Girders should be made as deep as the timber will allow : 
if their being increased in size should reduce the height of a story 
a few inches, it would be better than to have a house suffer from 
defective ceilings and insecure floors. In the following rules for 
the scantling of girders, they are supposed to be placed at 10 feet 
apart. 

Case 1.— To find the depth, when the breadth of the girder 
and the length of bearing are given. i?z/ie.— Divide the square 
of the length in feet, by the breadth in inches; and the cube-root 
of the quotient, multiplied by 4-2 for pine, or by 4*3 for oak, will 
give the depth required in inches. Example.— What should be 
the depth of a pine girder, having a length of 20 feet and a breadth 
of 13 inches ? The square of 20 is 400, which, divided by 13, 
gives 30-77; the cube-root of this is 312, which, multiplied by 
4-2, gives 13 inches, the depth required. 



FRAMING. 155 

Case 2. — To find the breadth, when the length of bearing and 
depth are given. Rule. — Divide the square of the length in feet 
by the cube of the depth in inches ; and the quotient, multiplied 
by 74 for pine, or by 82 for oak, will give the breadth in inches. 
Example. — What should be the breadth of a pine girder, having 
a length of 18 feet and a depth of 14 inches ? The square of 
the length in feet, 324, divided by the cube of the depth in 
inches, 2744, gives -118 ; and this, multiplied by 74, gives 8'73 
inches, the breadth required. 

278. — When the breadth of a girder is more than about 12 
inches, it is recommended to divide it by sawing from end to end, 
vertically through the middle, and then to bolt it together with 
the sawn sides outwards. This is not to strengthen the girder, 
as some have supposed, but to reduce the size of the timber, in 
order that it may dry sooner. The operation affords also an op- 
portunity to examine the heart of the stick — a necessary precau- 
tion ; as large trees are frequently in a state of decay at the heart, 
although outwardly they are seemingly sound. When the halves 
are bolted together, thin slips of wood should be inserted between 
them at the several points at which they are bolted, in order to 
leave sufficient space for the air to circulate between. This 
tends to prevent decay ; which will be found first at such parts 
as are not exactly tight, nor yet far enough apart to permit the 
escape of moisture. 

279. — When girders are required for a long bearing, it is usual 
to truss them ; that is, to insert between the halves two pieces of 
oak which are inclined towards each other, and which meet at 
the centre of the length of the girder, like the rafters of a roof- 
truss, though nearly if not quite concealed within the girder. 
This, and many similar methods, though extensively practised, 
are generally worse than useless ; since it has been ascertained 
that, in nearly all such cases, the operation has positively weak- 
ened the girder. 

A girder may be strengthened by mechanical contrivance, when 



156 AMERICAN HOUSE-CARPENTER, 




Fig. 205. 

its depth is required to be greater than any one piece of timber 
will allow. Fig' 205 shows a very simple yet scientific method 
of doing this. The two pieces of which the girder is composed 
are bolted, or pinned, together, having keys inserted between to 
prevent the pieces from sliding. The keys should be of hard 
wood, well seasoned. The two pieces should be about equal in 
depth, in order that the joint between them may be in the neutral 
line. (See Art. 254.) The thickness of the keys should be 
about half their breadth, and the amount of their united thick- 
nesses should be equal to a trifle over the depth and one-third of 
the depth of the girder. Instead of bolts or pins, iron hoops are 
sometimes used ; and when they can be procured, they are far 
preferable. In this case, the girder is diminished at the ends, 
and the hoops driven from each end towards the middle. 

280. — Beams may be spliced, if none of a sufficient length can 
be obtained, though not at or near the middle, if it can be avoided. 
(See Art. 259 and 332.) Girders should rest from 9 to 12 inches on 
the wall, and a space should be left for the air to circulate around 
the ends, that the dampness may evaporate. Floor-timbers are 
supported at their ends by walls of considerable height. They 
should not be permitted to rest upon intervening partitions, which 
are not likely to settle as much as the walls ; otherwise the une- 
qual settlements will derange the level of the floor. As all floors, 
however well-constructed, settle in some degree, it is advisable to 



FRAMING. 157 

frame the joists a little higher at the middle of the room than at 
its sides, — as also the ceiling-joists and cross-furring, when either 
are used. In single-joisted floors, for the same reason, the 
rounded edge of the stick, if it have one, should be placed up- 
per most. 

If the floor-plank are laid down tempoiarily at first, and left to 
season a few months before they are finally driven together and 
secured, the joints will remain much closer. But if the edges ot 
the plank are planed after the first laying, they will shrink again ; 
as it is the nature of wood to shrink after every planing however 
dry it may have been before. 

PARTITIONS. 

281. — Too little attention has been given to the construction of 
this part of the frame- work of a house. The settling of floors 
and the cracking of ceilings and walls, which disfigure to so great 
an extent the apartments of even our most costly houses, may be 
attributed almost solely to this negligence. A square of parti- 
tioning weighs about half a ton, a greater weight, when 
added to its customary load, such as furniture, storage, 
(fcc, than any ordinary floor is calculated to sustain. Hence 
the timbers bend, the ceilings and cornices crack, and the whole 
interior part of the house settles ; showing the necessity for 
providing adequate supports independent of the floor-timbers. 
A partition should, if practicable, be supported by the walls 
with which it is connected, in order, if the walls settle, that 
it may settle with them. This would prevent the separation of 
the plastering at the angles of rooms. For the same reason, a 
firm connection with the ceiling is an important object in the con- 
struction of a partition. 

282. — The joists in a partition should be so placed as to dis- 
charge the weight upon the points of support. All oblique pieces 
in a partition, that tend not to this object, are much better omitted. 
Fig. 206 represents a partition having a door in the middle. Its 



168 



AMERICAN H0USE-CARPE:NTER. 



1 


























•WW 






. J_ 




/. 


_ 


1 


^ 


1 

i 

1 










i 


1 


m 


f 


\ 


^ 


1 


% 


d-, 


M-N^ 


























tJ 



Fi£r. 206. 




\}.\\'i\ J 



Fijr. 207. 



construction is simple but effective. Fig- 207 shows the manner 
of constructing a partition having doors near the ends. The truss 
is formed above the door-heads, and the lower parts arc suspended 
from it. The posts, a and Z>, are halved, and nailed to the tie, c d, 
and the sill, ef. The braces in a trussed partition should be 
placed, so as to form, as near as possible, an angle of 40 degrees 
with the horizon. In partitions that are intended to support only 
their own weight, the principal timbers may be 3x4 inches for a 
20 feet span, 3ix5 for 30 feet, and 4x6 for 40. The thickness of 
the fiUing-in stuff may be regulated according to what is said at 
Art. 271, in regard to the width of furring for plastering. The 



FRAMING. 



159 



filling-in pieces should be stiffened at about every three feec by- 
short struts between. 

All superfluous timber, besides being an unnecessary load upon 
the points of support, tends to injure the stability of the plaster- 
ing ; for, as the strength of the plastering depends, in a great mea- 
sure, upon its clinch, formed by pressing the mortar through the 
space between the laths, the narrower the surface, therefore, upon 
which the laths are nailed, the less will be the quantity of plas- 
tering unclinched, and hence its greater security from fractures. 
For this reason, the principal timbers of the partition should have 
their edges reduced, by chamfering off the corners. 



cr 



(^ 



'/ 



■/ 



K 



^ 



di 



El 



^ 



!^ 



feJ 



Fiff. 2U8. 



283. — When the principal timbers of a partition require to be 
large for the purpose of greater strength, it is a good plan to omit 
the upright filling-in pieces, and in their stead, to place a few hori- 
zontal pieces ; in order, upon these and the principal timbers, to 
nail upright battens at the pi-oper distances for lathing, as in Fig. 
:208. A partition thus constructed requires a little more space 
than others ; but it has the advantage of insuring greater stability 
to the plastering, and also of preventing to a good degree the con- 
versation of one room from being heard in the other. When a 
partition is required to support, in addition to its own weight, that 
of a floor or some other burden resting upon it, the dimensionK. oi 



160 AMERICAN HOUSE-CARPENTER. 

the timbers ma^ be ascertained, by applying the principles which 
regulate the laws of pressure and those of the resistance of tim- 
ber, as explained at the lirst part of this section. The following 
data, however, may assist in calculating the amount of pressure 
upon partitions: 

284.— The weight of a square, (tliat is, a hundred square feet,) 
of partitioning maybe estimated at from 1500 to 2000 lbs.; a 
square of single-joisted flooring, at from 1200 to 2000 lbs. ; a 
square of framed flooring, at from 2700 to 4500 lbs.: and the 
weight of a square of deafenings (as described at the latter part 
of Art. 271,) at about 1500 lbs. 

When a floor is supported at two opposite extremiti(?s, and by a 
partition introduced midway, one-half of the weight of the whole 
floor will then be supported by the partition. As the settling of 
partitions and floors, which is so disastrous to plastering, is fre- 
quently owing to the shrinking of the timber and to ill-made 
joints, it is very important that the timber be seasoned and the 
work well executed. 

ROOFS.* 

285. — In ancient buildings, the Norman and the Gothic, the 
walls and buttresses were erected so massive and firm, that it was 
customary to construct their roofs without a tie-beam; the walls 
being abundantly capable of resisting the lateral pressure exerted 
by the rafters. But in modern buildings, the walls are so slightly 
built as to be incapable of resisting scarcely any oblique pressure ; 
and hence the necessity of constructing the roof so that all 
oblique and lateral strains may be removed; as, also, that instead 
of having a tendency to separate the walls, the roof may contri- 
bute to bind and steady them. 

286. — In estimating the pressures upon any certain roof, for the 
purpose of ascertaining the proper sizes for the timbers, calcula- 
tion must be made for the pressure exerted by the wind, and, if 

* See also Art. 228. 



FRAMING. 



161 



in a cold climatej for the weight of snow, in addition to the weight 
of the materials of which the roof is composed. The force of 
wind may be calculated at 40 lbs. on a square foot. The weight 
of snow will be of course according to the depth it acquires. 
{See weio-Jit of materials, in Appendix.) In a severe climate, 
roofs ought to be constructed steeper than in a milder one ; in order 
that the snow may have a tendency to slide off before it becomes of 
sufficient weight to endanger the safety of the roof. The inclina- 
tion should be regulated in accordance with the qualities of the 
material with which the roof is to be covered. The following table 
may be useful in determining the inclination, and in estimating 
the weight of the various kinds of covering : 



MATERIAL. 


INCLINATION. 


WEIGHT UPON A SQUARE FOOT. 


Tin, 


Rise 1 inch to a foot. 


f to l\ lbs. 


Copper, 


;; 1 " 


1 to 1^ " 


Lead, 


" 2 inches ** 


4 to 7 " 


Zinc, 


'• 3 " " 


li to 2 " 


Short pine shingles, 


'• 5 " " 


lito2^ " 


Loner cypress shingles. 


a Q u 


4 to 5 " 


Slate, 


u (3 u a 


5 to 9 " 



The weight of the covering, as above estimated, is that of the 
material only, added to the weight of whatever is used to fix it to 
the roof, such as nails, <fcc. ; what the material is laid on, such as 
plank, boards or lath, is not included. 

287. — Fig. 209 to 212 give a general idea of the usual manner 
of constructing trusses for roofs : c, {Fig. 209,) is a common 




162 



AMERICAN HOUSE-CARPENTEK. 




!< '^^ feet. 



Fig. 211. 




FRAMING. 163 

'after ; J? is a principal rafter ; A: is a king-post ; 5 is a strut ; S.. 
[Fig. 211,) is a straining-beam ; Q is a queen-post ; T is a tie- 
beam ; and P, P, {Fig: 212,) are purlins. In constructing a roof 
of importance, the trusses should be placed not over 10 feet apart, 
the principal rafter supported by a strut at every purlin, the purlin 
notched on instead of being framed into the principal rafters, and 
the tie-beam supported at proper distances, according to the weight 
of the ceiling or whatever else it is required to support. 

288. — The dimensions of the timbers may be found in accord- 
ance with the principles explained at the first part of this section ; 
but for general purposes, the following rules, deduced from the 
experience of practical builders and from scientific principles, 
may be found useful : these rules give the dimensions of the piece 
at its smallest part. 

289. — To find the dimensions of a king-post. Rule. — Mul- 
tiply the length of the post in feet by the span in feet. Then 
multiply this product by the decimal, 0*12, for pine, or by 0-13 
for oak, which will give the area of the king-post in inches : and 
divide this area by the breadth, and it will give the thickness ; or 
by the thickness for the breadth. Exainple. — What should be 
the dimensions of a pine king-post, 8 feet long, for a roof having 
a span of 25 feet? 8 times 25 is 200; this, multiplied by the 
decimal, 0*12, gives 24 inches for the area: 4x6, therefore, would 
be a good size at the smallest part. 

290. — To find the di?nensions of a queen-post. Rule. — Mul- 
tiply the length in feet, of the queen-post or suspending piece, by 
that part of the length of the tie-beam it supports, also in feet. 
This product, multiplied by the decimal, 0*27, for pine, or by 0*32 
for oak, will give the area of the post in inches ; and dividing 
this area by the thickness will give the breadth. Fxa??iple. — 
The queen-posts in Fig. 210 support each |- of the tie-beam, 
which is 12f feet. To make them of pine, 6 feet long, what 
should be their dimensions I 12f j multiplied by 6, gives 76, 



164 AMERICAN HOUSE-CARPENTER. 

which, multiplied by 0'2T, gives 20*52 ; which indicates a size of 
about 4x5^. 

291. — To fold the dimensions of a tie-beam, that is required 
to support a ceiling only. Rule. — Divide the length of the 
longest unsupported part by the cube-root of the breadth ; and the 
quotient, multiplied by 1*47 for pine, or by 1*52 for oak, will give 
the depth in inches. Example. — The length of the longest un- 
supported part of the tie-beam in Fig. 210 is 12f feet. What 
should be the depth of the tii^-luMm, the breadth being 6 inches, 
and the kind of wood, pine? The cube-root of 6 is 1*82, and 12f, 
divided by 1*82, gives a quotient of 6*956 ; this, multiplied by 
1*47, gives 10"225. The size of the tie-beam, therefore, maybe 
6x10^. When there are rooms in the roof, the dimensions for 
the tie-beam can be found by the rule for girders, {Art. 277.) 

292. — To find the dimeiisions of a principal rafter when 
there is a king-post in the middle. Rule. — Multiply the square 
of the length of the rafter in feet, by the span in feet ; and divide 
the product by the cube of the thickness in inches. For pine, 
multiply the quotient by '096, which will give the depth in 
inches. Example. — What should be the depth of a rafter of 
pine, 22'36 feet long, and 6 inches thick, the roof having a span 
of 40 feet ? The square of 22-36 is 500 nearly, this, multiplied by 
40, gives 20000 ; and this, divided by 216, the cube of the thick- 
ness, gives 92*59 ; which, multiplied by -096, equals 8*888. The 
size of the rafter should, therefore, be 6x8|. 

293. — To find the dimensions of a principal rafter when two 
queen-posts are used instead of a king-post. Rule. — The 
same as the last, except that the decimal, 0*155, must be used 
instead of '096. Example. — What should be the dimensions ot 
a principal rafter, having a length of 14 feet, (as in Fig. 210,) and 
a thickness of 6 inches, when the span of the roof is 38 feet 
and the wood is pine? The square of 14 is>196, which, multi- 
plied by 38, gives 74 18 ; this, divided by 216, the cube of 6, gives 



FRAMING. 165 

34 IS, which, multiplied by 0-155, gives 5*34. The size of the 
rafter should, therefore, be 6x5|. 

294. — To find the dimensions of a straining-beam. In or 
der that this beam may be the strongest possible, its depth should 
be to its thickness as 10 is to 7. Rule. — Multiply the square-root 
of the span in feet^ by the length of the straining-beam in feet, 
and extract the sqi^ere-root of the product. Multiply this root by 
()'9 for pine, which will give the depth in inches To find the 
thickness, multiply the depth by the decimal, 0"7. Example. — 
What should be the dimensions of a pine straining-beam, 12 feet 
long, for a span of 38 feet ? The square-root of the span is 6*164, 
which, multiplied by 12, gives 73*968 ; the square-root of this is 
nearly 8*60, which, multiplied by 0*9, gives 7*74 — the depth. 
This, multiplied by 0*7, gives 5*418 — the thickness. Therefore, 
the beam should be 5|x7|, or 5|x8. 

295. — To find the dim^ensions of stmts and braces. Rule. — 
Multiply the square-root of the length supported in feet, by the 
length of the brace or strut in feet ; and the square-root of the 
product, multiplied by 0*8 for pine, will give the depth in inches ; 
and the depth, multiplied by the decimal, 0*6, will give the thick- 
ness in inches. Example. — In Fig. 210, the part supported by 
the brace or strut, a, is equal to half the length of the principal 
rafter, or 7 feet ; and the length of the brace is 6 feet : what 
should be the size of a pine brace ? The square-root of 7 is 2*65, 
which, multiplied by 6, gives 15-9 ; the square-root of this is 3'99, 
which, multiplied by 0*8, gives 3*192 — the depth. This, multi- 
plied by 0*6, gives 1*9152, the thickness. Therefore, the brace 
should be 2x3 inches. 

It is customary to make the principal rafters, tie-beam, posts 
and braces, all of the same thickness, that the whole truss may 
be of the same thickness throughout. 

296. — To find the dim,ensions of purlins. Rule. — Multiply 
the cube of the length of the purlin in feet, by the distance the 
purlins are apart in feet ; and the fourth root of the product for 
pine will give the depth in inches ; or multiply by 1*04 to obtain 



166 



AMERICAN HOUSE-CARPENTER. 



the depth for oak ; and the depth, multiplied by the decimal, 0*6, 
will give the thickness. Example. — What should be the dimen- 
sions of pine purlins, 9 feet long and 6 feet apart ? The cube of 
9 is 729, which, multiplied by 6, gives 4374; the fourth root of 
this is 8-13 — the required depth. This, multiplied by 0-6, gives 
4*87S — the thickness. A proper size for them would be about 
5x8 inches. Purlins should be long enough to extend over two, 
three or more trusses. 

297. — To find the dbnensions of co9nmon rafters. The fol- 
lowing rule is for slate roofs, having the rafters placed 12 inches 
apart. Shingle roofs may have rafters placed 2 feet apart. The 
dimensions of rafters for other kinds of covering may be found by 
reference to the table at Art. 286. and the laws of pressure at the 
first part of this section. Rule. — Divide the length of bearing in 
feet, by the cube-root of the breadth in inches ; and the quotient, 
multiplied by 0-72 for pine, or 0'74 for oak, will give the depth in 
inches. Exaiiiple. — What should be the depth of a pine rafter, 
7 feet long and 2 inches thick ? 7 feet, divided by 1*26, the cube- 
foot of 2, gives 5*55, which, multiplied by 0'72, gives nearly 4 
inches — the depth required. 

298. — If, instead of framing the principal rafters and straining- 
beam into the king ;md the queen posts, they be permitted to abut 
against each other, and the king and the queen posts be made in 
halves, notched on and bolted, or strapped to each other and to the 
tie-beam, much of the ill effects of shrinking in the heads of the 
king and the queen posts will be avoided. (See Art. 339 and 340.) 




FRAMING. 



167 



299. — Fig. 213 shows a method of constructing a truss having 
a huilt-rih in the place of principal rafters. The proper form 
for the curve is that of a parabola, {Art. 127.) This curve, when 
as flat as is described in the figure, approximates so near to that of 
the circle, that the latter may be used in its stead. The height, 
a 6, is just half of a c, the curve to pass through the middle of 
the rib. The rib is composed of two series of abutting pieces, 
bolted together. These pieces should be as long as the dimen- 
sions of the timber will admit, in order that there may be but few 
joints. The suspending pieces are in halves, notched and bolted 
to the tie-beam and rib, and a purlin is framed upon the upper end 
of each. A truss of this construction needs, for ordinary roofs, 
no diagonal braces between the suspending pieces, but if extra 
strength is required the braces may be added. The best place 
for the suspending pieces is at the joints of the rib. A rib of this 
kind will be sufliciently strong, if the area of its section contain 
about one-fourth more timber, than is required for that of a strain- 
ing-beam for a roof of the same size. The proportion of the 
depth to the thickness should be about as 10 is to 7. 




Fig. 214. 



300. — Some writers have given designs for roofs similar to Fig. 
214, having the tie-beam omitted for the accommodation of an 
arch in the ceiling. This and all similar designs are seriously 
objectionable, and should always be avoided ; as the small height 
gained by the omission of the tie-beam can never compensate for 
the powerful lateral strains, which are exerted by the oblique posi- 
tion of the supports, tending to separate the walls. Where an arch 



168 



AMERICAN HOUSE-CARPENTER. 



is required in the ceiling, the best plan is to carry up the walls 
as high as the top of the arch. Then, by using a horizontal tie- 
beam, the oblique strains will be entirely removed. Many a pub- 
lic building in this place and vicinity, has been all but ruined by 
the settling of the roof, consequent upon a defective plan in the 
formation of the truss in this respect. . It is very necessary, there- 
fore, that the horizontal tie-beam be used, except where the walls 
are made so strong and firm by abutments, or other support, as to 
prevent a possibility of their separating. 




FliT. 2iJ. 



301. — Fig' 215 is a meihod of obtaining the proper lengths and 
bevils for rafters in a hip-roof, a b and b c are walls at the angle 
of the building : & e is the seat of the hip-rafter and g f of a. 
jack or cripple rafter. Draw e h, at right angles to b e, and make 
it equal to the rise of the roof; join b and h, and h b will be the 
length of the hip-rafter. Through e, draw d i, at right angles 
to 6 c; upon 6, with the radius, b A, describe the arc, h i, cutting 
d i'mi ; join b and i, and extend gf to meet biin j ; then gj will 



FRAMING. 



169 



be the length of the jack-rafter. The length of each jack-rafter is 
found in the same manner — by extending its seat to cut the line, 
b i. From/, draw fk, at right angles tofg, also//, at right 
angles to be; make/ A: equal to /Z by the arc, I k, or make g- k 
equal to gj by the arc, J k ; then the angle atj will be the top- 
bevil of the jack-rafters, and the one at k will be the down-bevil.^ 
302. — To find the backing of the hip-rafter. At any con- 
venient place in b e, {Fig.. 215,) as o, draw m n^ at right angles to 
be; from o, tangical to b A, describe a semi-circle, cutting b e in 
5 ; join ni and s and n and s ; then these lines will form at 5 the 
proper angle for beviling the top of the hip-rafter. 

DOMES.t 




Fig. 216. 




Fig. 217. 

♦ The lengths and bevils of rafters for roof-valleys can also be found by the abov? 
process t See also Art. 227 

22 



170 



AMERICAN HOUSE-CARPENTER. 



303. — The most usual form for domes is that of the sphere, the 
base being circular. When the interior dome does not rise too 
high, a horizontal tie may be thrown across, by which any de- 
gree of strength required may be obtained. Fig. 216 shows a 
section, and Fig. 217 the plan, of a dome of this kind, a h being 
the tie-beam in both. Two trusses of this kind, [Fig. 216,) pa- 
rallel to each other, are to be placed one on each side of the open- 
ing in the top of the dome. Upon these the whole framework is to 
depend for support, and their strength must be calculated accord- 
ingly. (See the first part of this section, and Art. 286.) If the 
dome is large and of importance, two other trusses may be intro- 
duced at right angles to the foregoing, the tie-beams being pre- 
served in one continuous length by framing them high enough to 
pass over the others. 




Fiff. 918. 




Fig. 219. 



304. — When the interior dome rises too high to admit of a level 



FRAMING. 171 

tie-beam, the framing may be composed of a succession of ribs 
standing upon a continuous circular curb of timber, as seen at 
Fig. 218 and 219,— the latter being a plan and the former a sec- 
tion. This curb must be well secured, as it serves in the place 
of a tie-beam to resist the lateral thrust of the ribs. In small 
domes, these ribs may be easily cut from wide plank ; but, where 
an extensive structure is required, they must be built in two 
thicknesses so as to break joints, in the same manner as is descri- 
bed for a roof at Art. 299. They should be placed at about two 
feet apart at the base, and strutted as at a in Fig. 218. 

305. — The scantling of each thickness of the rib may be as 
follows : 

For domes of 24 feet diameter, 1x8 inches. 
" '• 36 " lixlO " 

" ' 60 '' 2x13 " 

" " 90 " 2|xl3 " 

" " 108 " 3x13 " 

306. — Although the outer and the inner surfaces of a dome 
may be finished to any curve tliat may be desired, yet the framing 
should be constructed of such a form, as to insure that the curve 
of eqiiilibriiun will pass through the middle of the depth of the 
framing. The nature of this curve is such that, if an arch or 
dome be constructed in accordance with it, no one part of the 
structure will be less capable than another of resisting the strains 
and pressures to whicli the v/hole fabric may be exposed. Tiie 
curve of equilibrium for an arched vault or a roof, where the load 
is equally diffused over the whole surface, is that of a parabola, 
[Art. 127' ;) for a dome, having no Icudern^ tower or cupola above 
it, a aihic parabola, {Fig. 220 ;) and for one having a tower, &c., 
above it, a curve approaching that of an hyperbola must be adopted, 
as the greatest strength is required at its upper parts. If the 
curve of a dome be circular, (as in the vertical section. Fig. 218,) 
ihe pressure will have a tendency to burst the dome outwards at 
-bout one-third of its height. Therefore, when this form is used 



172 



AMERICAN HOUSE-CARPENTER. 



in the construction of an extensive dome, an iron band should be 
placed around the framework at that height ; and whatever may- 
be the form of the curve, a band or tie of some kind is necessary 
around or across the base. 

If the framing be of a form less convex than the curve of 
equilibrium, the weight will have a tendency to crush the ribs in- 
wards, but this pressure may be effectually overcome by strutting 
between the ribs ; and hence it is important that the struts be so 
placed as to form continuous horizontal circles. 








^ 










/ 






/ 










J 


/ 










Oj 


/ 












/ 














A 















a j I h 



uo. 



307. — To describe a cubic parabola. Let a b, {Fig. 220,) be 
the base and b c the height. Bisect a b at d, and divide a d into 
100 equal parts; of these give d e 26, ef lSi,f g 14|, g h 12^, 
h i lOf, ij 9i, and the balance, 8|, to ; a; divide b c into 8 equcil 
parts, and, from the points of division, draw lines parallel to a 6, 
to meet perpendiculars from the several points of division in a b, 
at the points, o, o, o, ifcc. Then a curve traced through these 
points will be the one required. 

308.— Small domes to light stairways, &c., are frequently made 
elliptical in both plan and section ; and as no two of the ribs in 
one quarter of the dome are alike in form, a method for obtaining 
the curves is necessary. 

309.— To find the curves for the ribs of an elliptical dome. 
Let a 6 c d, [Fig. 221,) be the plan ot a dome, and e f the seat 



FRAMING. 



173 




Fig. 2-1 



of one of the ribs. Then take e f for the transverse axis and 
twice the rise, o g, of the dome for the conjugate, and describe, 
(according to Art. 115, 116, &c.,) the semi-ellipse, e g f, which 
will be the curve required for the rib, e g f. The other ribs are 
found in the same manner. 



b 4 




Fig. 222. 



310. — To find the shape of the covering for a spherical 
dome. Let A, (Fig. 222,) be the plan and B the section of a 
given dome. From a, draw a c, at right angles to a b ; find the 
stretch-out, {Art, 92,) of o b, and make d c equal to it ; divide the 
arc, 6, and the line, d c, each into a like number of equal parts, 



174 



AMERICAN HOUSE-CARPENTER. 



as 5, (a large number will insure greater accuracy than a small 
one ;) uponc, through the several points of division in c d^ describe 
the arcs, o c? o, 1 e 1, 2/ 2, &c. ; make d o equal to half the width 
of one of the boards, and draw o s, parallel to a c ; join s and a, 
and from the points of division in the arc, o b, drop perpendicu- 
lars, meeting a 5 in ij k I ; from these points, draw i 4, j 3, (fee, 
parallel to a c; make d o, e l,&c., on the lower side of a c, equal 
to d 0, e 1, &c., on the upper side ; trace a curve through the 
points, 0, 1, 2, 3, 4, c, on each side o{ d c ; then o c o will be 
the proper shape for the board. By dividing the circumference of 
the base. A, into equal parts, and making the bottom, o d o,o{ the 
board of a size equal to one of those parts, every board may be 
made of the same size. In the same manner as the above, the 
shape of the covering for sections of another form may be found, 
such as an ogee, cove, &c. 




311. — Tojiiid the ciirve of the hoards when laid in horizon- 
tal cottrses. Let ABC, {Fig. 223,) be the section of a given 
dome, and D B its axis. Divide B C into as many parts as 
there are to be courses of boards, in the points, 1, 2, 3, Sec. ; 
through 1 and 2, draw a line to meet the axis extended at a ; 
then a will be the centre for describing the edges of the board, F. 
Through 3 and 2, draw 36; then b will be the centre for describing 
F. Through 4 and 3, draw Ad; then d will be the centre for G 
B is the centre for the arc, 1 o. If this method is taken to find 



FRAMING. 



175 



the centres for the boards at the base of the dome, they would 
occur so distant as to make it impracticable : the following method 
is preferable for this purpose. G being the last board obtained by 
the above method, extend the curve of its inner edge until it 
meets the axis, D B, in e ; from 3, through e, draw 3 /, meeting 
the arc, A B, in/; join /and 4, /and 5 and/ and 6, cutting the 
axis, D B, in s, n and ?n ; from 4, 5 and 6, draw lines parallel to 
A Cand cutting the axis in c, p and r; make c 4, {Fig, 224,) 




equal to c 4 in the preyious figure, and c s equal to c 5 also in the 
previous figure ; then describe the inner edge of the board, H^ 
according to Art. 87 : the outer edge can be obtained by gauging 
from the inner edge. In like manner proceed to obtain the next 
board — taking p .5 for half the chord and p n for the height of the 
segment. Should the segment be too large to be described 
easily, reduce it by finding intermediate points in the curve, as at 
Art. 86. 




312. — To find the shape of the angle-rib for a polygonal 
dome. Let AG H^ {Fig. 225,) be the plan of a given dome, and 



176 



AMERICAN HOUSE-CARPENTER. 



O Z> a vertical section taken at the line, e f. From 1, 2, 3, (fee, 
in the arc, C JD, draw ordinates, parallel to A Z>, to meet/ G ; 
from the points of intersection on / G^ draw ordinates at right- 
angles to/ G ; make s 1 equal to o 1, 5 2 equal to 2, &c. ; ^hen 
GfB^ obtained in this way, will be the angle-rib required. The 
best position for the sheathing-boards for a dome of this kind is 
horizontal, but if they are required to be bent from the base to 
the vertex, their shape may be found in a similar manner to that 
shown at Fig. 222. 

BRIDGES. 

313. — Yarious plans have been adopted for the construction of 
bridges, of which perhaps the following are the most useful. 
Fig. 22G shows a method of constructing wooden bridges, where 
the banks of the river are higli enough to permit the use of the 
tie-beam, a b. The upright pieces, c (/, are notched and bolted 
on ni pairs, for the support of the tie-beam. A bridge of this 
construction exerts no lateral pressure upon the abutments. This 
method maybe employed even where the banks of the river are 
low, by letting the timbers for the roadway rest immediately upon 
the tie-beam. In this case, the framework above will serve the 
purpose of a railing. 




Fig. 226. 



314. — Fig. 227 exhibits a wooden bridge without a tie-beam. 
Where staunch buttresses can be obtained, this method may be 
recommended ; but if there is any doubt of their stability, it 



FRAMING. 



177 




Fig. 227. 



should not be attempted, as it is evident that such a system of 
framing is capable of a tremendous lateral thrust. 




Fig. 22a 



315. — Fig. 228 represents a wooden bridge in which a built-rib, 
(see Art. 299,) is introduced as a chief support. The curve of 
equilibrium will not differ much from that of a parabola : this, 
therefore, may be used — especially if the rib is made gradually a 
little stronger as it approaches the buttresses. As it is desirable 
that a bridge be kept low, the following table is given to show the 
least rise that may be given to the rib. 



Span in feet. 


Least rise in feet. 


Span in feet 


Least rise in feet. 

7 1 


Span in feet. 


Least rise in feet. 


30 


0-5 


120 


280 


24 


40 


0-8 


140 


8 


300 


28 


50 


1-4 


160 


10 


320 


32 


60 


2 


180 


11 


350 


39 


70 


2i 


200 


12 


380 


47 


80 


3 


220 


14 


400 


53 


90 


4 


240 


17 






100 


5 


260 


20 







The rise should never be made less than this, but in all cases 

23 



L78 



AMERICAN HOUSE-CARPENTER. 



greater if practicable ; as a small rise requires a greater quantity 
of timber to make the bridge equally strong. The greatest uni- 
form weight with which a bridge is likely to be loaded is, proba- 
bly, that of a dense crowd of people. This may be estimated at 
120 pounds per square foot, and the framing and gravelled road- 
way at 180 pounds more ; which amounts to 300 pounds on a 
square foot. The following rule, based upon this estimate, may 
be useful in determining the area of the ribs. Rule. — Multiply 
the width of the bridge by the square of half the span, both in 
feet ; and divide this product by the rise in feet, multiplied by the 
number of ribs ; the quotient, multiplied by the decimal, 
O'OOll, will give the area of each rib in feet. When the road- 
way is only planked, use the decimal, 0*0007, instead ot 
O'OOll. Example. — What should be the area of the ribs for a 
bridge of 200 feet span, to rise 15 feet, and be 30 feet wide, with 
3 curved ribs ? The half of the span is 100 and its square is 
10,000 ; this, multiplied by 30, gives 300,000, and 15, multi- 
plied by 3, gives 45 ; then 300,000, divided by 45, gives 6666f , 
which, multiplied by 0-0011, gives 7*333 feet, or 1056 inches for 
the area of each rib. Such a rib may be 24 inches thick by 44 
inches deep, and composed of 6 pieces, 2 in width and 3 in depth. 




Fig. 229. 



316. — The above rule gives the area of a rib, that would be re- 
quisite to support the greatest possible uniform load. But in 
large bridges, a variable load, such as a heavy wagon, is capable 
of exerting much greater strains ; in such cases, therefore, the 
rib should be made larger. The greatest concentrated load a 



FRAMING. 179 

bridge will be likely to encounter, may be estimated at from about 
20 to 50 thousand pounds, according to the size of the bridge. 
This is capable of exerting the greatest strain, when placed at 
about one-third of the span from one of the abutments, as at b. 
{Fig. 229.) The weakest point of the segment, b g- c, is at g^ 
the most distant point from the chord line. The pressure exerted 
at b by the above weigl^t, may be considered to be in the direction 
of the chord lines, b a and be; then, by constructing the paral- 
lelogram of forces, e b f d, according to Art. 248, b f will show 
the pressure in the direction, b c. Then the scantling for the rib 
may be found by the following rule. 

Rule. — Multiply the pressure in pounds in the direction, b c. 
by the decimal, 0*0016, for white pine, 0*0021 for pitch pine, and 
0'0015 for oak, and the product by the decimal representing the 
sine of the angle, g b h, to a radius of unity. Divide this pro- 
duct by the united breadth in inches of the several ribs, and the 
cube-root of the quotient, multiplied by the distance, b c, in feet. 
will give the depth of the rib. Example. — In a bridge of 200 
feet span, 15 feet rise, having 3 ribs each 24 inches thick, or 72 
inches whole thickness, the pressure in the direction, 6 c, is found 
to be 166,000 lbs., and the sine of the angle, g b h, is 0*1 — what 
should be the depth of the rib for white pine? 166,000, mul- 
tiplied by 0-0016, gives 265*6, which, multiplied by 0*1, gives 
26-56 ; this, divided by 72, gives 0-3689. The cube-root of the 
last sum is 0-717 nearly, and the distance, b c, is 135 feet : then, 
0-'^ 17, multiplied by 135, gives 96f inches, the depth required. 
By this, each rib will require to be 24x97 inches, in order to en- 
counter without injury the greatest possible load. 

317. — In constructing these ribs, if the span be not over 50 
feet, each rib may be made in two or three thicknesses of timber, 
(three thicknesses is preferable,) of convenient lengths bolted 
together ; but, in larger spems, where the rib will be such as to 
render it difficult to procure timber of sufficient breadth, they 
may be constructed by bending the pieces to the proper curv€^ 



)80 



AMERICAN HOUSE-CARPENTER. 



and bolting them together. In this case, where timber of suffi- 
cient length to span the opening cannot be obtained, and scarfing 
is necessary, such joints must be made as will resist both tension 
and compression, (see Fig. 238.) To ascertain the greatest depth 
for the pieces which compose the rib, so that the process of bend- 
ing may not injure their elasticity, multiply the radius of curvature 
in feet by the decimal, 0*05, and the product will be the depth m 
inches. Example. — Suppose the curve of the rib to be described 
with a radius of 100 feet, then what should be the depth ? The 
radius in feet, 100, multiplied by 0*05, gives a product of 5 inches. 
White pine or oak timber, 5 inches thick, would freely bend to 
the above curve ; and, if the required depth of such a rib be 2( 
inches, it would have to be composed of at least 4 pieces. Pitch 
pine is not quite so elastic as white pine or oak — its thickness 
may be found by using the decimal, 0*046, instead of 0'05. 




Fig. 230. 



318. — When the span is over 250 feet, ?l framed rib, formed as 
in Fig. 230, would be preferable to the foregoing. Of this, the 
upper and the lower edges are formed as just described, by bend- 
ing the timber to the proper curve. The pieces that tend to the 
centre of the curve, called radials^ are notched and bolted on in 
pairs, and the cross-braces are halved together in the middle, and 
abut end to end between the radials. The distance between the 
ribs of a bridge should not exceed about 8 feet. The roadway 



FRAMING. 18i 

should be supported by vertical standards bolted to the ribs at 
about every 10 to 15 feet. At the place where they rest on the 
ribs, a double, horizontal tie should be notched and bolted on the 
back of the ribs, and also another on the under side ; and diago- 
na' braces should be framed between the standards, over the space 
between the ribs, to prevent lateral motion. The timbers for the 
roadway may be as light as their situation will admit, as all use- 
less timber is only an unnecessary load upon the arch. 

319. — It is found that if a roadway be 18 feet wide, two car- 
riages can pass one another without inconvenience. Its width, 
therefore, should be either 9, 18, 27 or 36 feet, according to the 
amount of travel. The width of the foot-path should be 2 feet 
for every person. When a stream of water has a rapid current, 
as few piers as practicable should be allowed to obstruct its 
course ; otherwise the bridge will be liable to be swept away by 
freshets. When the span is not over 300 feet, and the banks of 
the river are of sufficient height to admit of it, only one arch 
should be employed. The rise of the arch is limited by the form 
of the roadway, and by the height of the banks of the river 
(See Art. 315.) The rise of the roadway should not exceed one 
in 24 feet, but, as the framing settles about one in 72, the roadway 
should be framed to rise one in 18, that it may be one in 24 after 
settling. The commencement of the arch at the abutments — the 
springs as it is termed, should not be below high-water mark : 
and the bridge should be placed at right angles with the course of 
the current. 

320. — The best material for the abutments and piers of a 
bridge, is stone ; and, if possible, stone should be procured for the 
purpose. The following rule is to determine the extent of the 
abutments, they being rectangular, and built with stone weighing 
120 lbs. to a cubic-foot. Rule. — Multiply the square of the 
height of the abutment by 160, and divide this product by the 
weight of a square foot of the arch, and by the rise of the arch ; 
add unity to the quotient, and extract the square-root. Diminish 
the square-root by unity, and multiply the root, so diminished, by 



AMERICAN HOUSE-CARPENTER. 

half the span of the arch, and by the weight of a square-foot ot 
the arch. Divide the last product by 120 times the height of the 
abutment, and the quotient will be the thickness of the abutment. 
Exain'ple. — Let the height of the abutment from the base to the 
springing of the arch be 20 feet, half the span 100 feet, the weight 
of a square foot of the arch, including the greatest possible load 
upon it, 300 pounds, and the rise of the arch 18 feet — what should 
be its thickness ? The square of the height of the abutment, 
400, multiplied by 100, gives 64,000, and 300 by 18, gives 5400 ; 
64,000, divided by .5400, gives a quotient of 11*852, one added to 
this makes 12*852, the square-root of which is 3*6 ; this, less one, 
is 2*6 ; this, multiplied by 100, gives 260, and this again by 300, 
gives 78,000 ; this, divided by 120 times the height of the abut- 
ment, 2400, gives 32 feet 6 inches, the thickness required. 

The dimensions of a pier will be found by the same rule. 
For, although the thrust of an arch may be balanced by an ad- 
joining arch, when the bridge is finished, and while it remains 
uninjured ; yet, during the erection, and in the event of one arch 
being destroyed, the pier should be capable of sustaining the en- 
tire thrust of the other. 

321. — Piers are sometimes constructed of timber, their princi- 
pal strength depending on piles driven into the earth, but such 
piers should never be adopted where it is possible to avoid them ; 
for, being alternately wet and dry, they decay much sooner than 
the upper parts of the bridge. Spruce and elm are considered 
good for piles. Where the height from the bottom of the 
river to the roadway is great, it is a good plan to cut them off at 
a little below low-water mark, cap them with a horizontal tie, 
and upon this erect the posts for the support of the roadway. 
This method cuts oif the part that is continually wet from that 
which is only occasionally so, and thus affords an opportunity for 
replacing the upper part. The pieces which are immersed will 
last a great length of time, especially when of elm ; for it is a 
well-established fact, that timber is less durable when subject to 



FRAMING. 



183 



alternate dryness and moisture, than when it is either continually 
wet or continually dry. It has been ascertained that the piles 
under London bridge, after having been driven about 600 years, 
vere not materially decayed. These piles are chiefly of elm, and 
vholly immersed. 




322. — Centres for stone bridges. Fig- 231 is a design for a 
centre for a stone bridge where intermediate supports, as piles 
driven into the bed of the river, are practicable. Its timbers are 
so distributed as to sustain the weight of the arch-stones as they 
are being laid, without destroying the original form of the centre ; 
and also to prevent its destruction or settlement, should any of the 
piles be swept away. The most usual error in badly-constructed 
centres is, that the timbers are disposed so as to cause the framing 
to rise at the crown, during the laying of the arch-stones up the 
sides. To remedy this evil, some have loaded the crown with 
heavy stones ; but a centre properly constructed will need no 
such precaution. 

Experiments have shown that an arch-stone does not press 
upon the centring, until its bed is inclined to the horizon at an 
angle of from 30 to 45 degrees, according to the hardness of the 
stone, and whether it is laid in mortar or not. For general pur- 
poses, the point at which the pressure commences, may be con- 
sidered to be at that joint which forms an angle of 32 degrees 
with the horizon. At this pumt, the pressure is inconsiderable, 



184 AMERICAN HOUSE-CARPENTER. 

but gradually increases towards the crown. At an angle of 45 
degrees, the pressure equals about one-quarter the weight of the 
stone ; at 67 degrees, half the weight ; and when a vertical line, 
as a 6, {Fig. 232,) passing through the centre of gravity of 




Fig. 232. 



the arch-stone, does not fall within its bed, c d, the pressure may 
be considered equal to the whole weight of the stone. This will 
be the case at about 60 degrees, when the depth of the stone is 
double its breadth- The direction of these pressures is consid- 
ered in a line with the radius of the curve. The weight upon a 
centre being known, the pressure may be estimated and the tim- 
ber calculated accordingly. But it must be remembered that the 
whole weight is never placed upon the framing at once — as seems 
to have been the idea had in view by the designers of some cen- 
tres. In building the arch, it should be commenced at each but- 
tress at the same time, (as is generally the case,) and each side 
should progress equally towards the crown. In designing the 
framing, the effect produced by each successive layer of stone 
should be considered. The pressure of the stones upon one side 
should, by the arrangement of the struts, be counterpoised by that 
of the stones upon the other side. 

323. — Over a riv^cr whose stream is rapid, or where it is ne 
cessary to preserve an uninterrupted passage for the purposes of 
navigation, the centre must be constructed without intermediate 
supports, and without a continued horizontal tie at the base ; such 
a centre is shown at Fig. 233. In laying the stones from the 
base up to a and c, the pieces, b d and h d, act as ties to prevent 
any rising at b. After this, while the stones are being laid from 
a and from c to b, they act as struts : the piece, /^, is added foi 



185 




Fig. 233. 



additional security. Upon this plan, with some variation to suit 
circumstances, centres may be constructed for any span usual in 
stone-bridge building. 

324. — In bridge centres, the principal timbers should abut, and 
not be intercepted by a suspension or radial piece between. 
These should be in halves, notched on each side and bolted. 
The timbers should intersect as little as possible, for the more 
joints the greater is the settling ; and halving them togetlier is a 
bad practice, as it destroys nearly one-half the strength of the 
timber. Ties should be introduced across, especially where many 
timbers meet ; and as the centre is to serve but a temporary pur- 
pose, the whole should be designed with a view to employ the 
timber afterwards for other uses. For this reason, all unneces- 
sary cutting should be avoided. 

325. — Centres should be sufficiently strong to preserve a 
staunch and steady form during the whole process of building; 
for any shaking or trembling will have a tendency to prevent the 
mortar or cement from settijig. For this purpose, also, the cen- 
tre should be lowered a trifle immediately after the key-stone is 
laid, in order that the stones may take their bearing before the 
mortar is set : otherwise the joints will open on the under side. 
The trusses, in centring, are placed at the distance of from 4 to 
6 feet apart according to their strength and the weight of the 

24 



186 AMERICAN HOUSE-CARPENTER. 

arch. Between every two trusses, diagonal braces should be h* 
troduced to prevent lateral motion. 

326. — In order that the centre maybe easily lowered, the frames, 
or trusses, should be placed upon wedge-formed sills ; as is shown 
at dj {Fig. 233.) These are contrived so as to admit of the settling 
of the frame by driving the wedge, d, with a maul, or, in large 
centres, a piece of timber mounted as a battering-ram. The 
operation of lowering a centre should be very slowly performed, 
in order that the parts of the arch may take their bearing uni- 
formly. The wedge pieces, instead of being placed parallel with 
the truss, are sometimes made sufficiently long and laid through 
the arch, in a direction at right angles to that shown at Fig. 233. 
This method obviates the necessity of stationing men beneath the 
arch during the process of lowering ; and was originally adopted 
with success soon after the occurrence of an accident, in lower- 
ing a centre, by which nine men were killed. 

327. — To give some idea of the manner of estimating the 
pressures, in order to select timber of the proper scantling, calcu- 
late the pressure of the arch-stones from i to b, {Fig. 233,) and 
suppose half this pressure concentrated at a, and acting in the 
direction, a f. Then, by reference to the laws of pressure and 
the resistance of timber at Art. 248, 260, <fec., the scantlings of 
the several pieces composing the frame, b d a, may be computed. 
Again, calculate the pressure of that portion of the arch in^l;ded 
between a and c, and consider half of it collected at 6, and acting 
in a vertical direction ; then the amount of pressure on the beams, 
b d and b d, may be found by reference to the first part of this 
section, as above. Add the pressure of that portion of the arch 
which is included between i and b to half the weight of the cen- 
tre, and consider this amount concentrated at d, and acting in a 
vertical direction ; then, by constructing the parallelogram ot 
forces, the pressure upon dj may be ascertained* 

328. — As a short rule for calculating the scantlings of the tim- 
bers, let every strut be sufficiently braced, so tiiat it will yield Ic 



FRAMING. 187 

crushing before it will bend under the pressure — {A rt. 261.) Then 
divide the pressure in pounds by 1000, and the quotient will be 
the area of the strut in inches. For example, let the pressure 
upon a" strut, in the direction of its axis, be 60,000 lbs. This, 
divided by 1000, gives 60, the area of the strut in inches ; the 
size of the strut, therefore, might be 6x10. This rule is based 
upon experiments by which it has been ascertained, that 1000 
pounds is the greatest load that can be trusted upon a square inch 
of timber, without more indentation than would be compatible 
with the stability of the framing. The area ascertained by the 
rule, therefore, must have reference to the actual amount of sur- 
face upon which the load bears ; and should the strut have a tenon 
on the end, the area of tlie shoulders, instead of a section of the 
whole piece, must be equal to the amount given by the rule. 

329. — In the construction of arches, the voussolrs, or arch- 
stones, are so shaped that the joints between them are perpen- 
dicular to the curve of the arch, or to its tangent at the point at 
which the joint intersects the curve. In a circular arch, the 
■■oints tend toward the centre of the circle : in an elliptical 
arch, the joints may be found by the following process : 




330. — To find the direct iou of the joints for an elliptical 
arch. A joint being wanted at «, [Fig. 234,) draw lines from 
that point to the foci, /and/; bisect the angle, /a/ with the 
line, ah ; then a b will be the direction of the joint. 

331. — To find the direction of the joints for a parabolic arch. 
A joint being wanted at a, [Pig. 235,) draw a e, at right angles to 
the axis, eg; make c g equal to c e, and join a and g ; draw a A, at 
right angles to a ^ ; then a h will be the direction of the joint 



188 



AMERICAN HOUSE-CARPENTER. 





g 

f 


^/L/^ 


e \ 


/ ^ 


\ 



Fig. 235. 

The direction of the joint from h is found in the same mannei . 
The hnes, a g and 6/, are tangents to the curve at those pomts 
respectively ; and any number of joints in the curve may be ob- 
tained, by first ascertaining the tangents, and then drawing lines 
at right angles to them. 

JOINTS. 



Fis. 236. 



332. — Fig' 236 shows a simple and quite strong method ot 
lengthening a tie-beam ; but the strength consists wholly in the 
bolts, and in the friction of the parts produced by screwing the 
pieces firmly together. Should the timber shrink to even a small 
degree, the strength would depend altogether on the bolts. It 
would be made much stronger by indenting the pieces together ; 
as at the upper edge of the tie-beam in Fig. 237 ; or by placing 



^ 



-[z^ 



■ G^ 

FiL'. 237. 



keys in the joints, as at the lower edge in the same figure. This 
process, however, weakens the beam in proportion to the depth 
of the indents. 

333. — Fig. 238 shows a method of scarfing, or splicing, a tie- 
beam without bolts. The keys are to be of well-seasoned, hard 



FRAMING. 189 



Fii 



wood, andj if possible, very cross-grained. The addition of bolts 
would make this a very strong splice, or even white-oak pins 
would add materially to its strength. 



Fiff. 239. 



334. — Fig. 239 shows about as strong a splice, perhaps, as 
can well be made. It is to be recommended for its simplicity ; 
as, on account of their being no oblique joints in it, it can be 
readily and accurately executed. A complicated joint is the 
worst that can be adopted ; still, some have proposed joints that 
seem to have little else besides complication to recommend 
them. 

335. — In proportioning the parts of these scarfs, the depths of 
all the indents taken together should be equal to one-third of the 
depth of the beam. In oak, ash or elm, the whole length of the 
scarf should be six times the depth, or thickness, of the beam, 
when ihere are no bolts ; but, if bolts instead of indents are used, 
then three times the breadth ; and, when both methods are com- 
bined, twice the depth of the beam. The length of the scarf in 
pme and similar soft woods, depending wholly on indents, should 
be about 1-2 times the thickness, or depth, of the beam ; when 
depending wholly on bolts, 6 times the breadth ; and, when both 
methods are combined, 4 times the depth. 

1 — — — - — — -- — -x 

Fig. 240. 

336. — Sometimes beams have to be pieced that are required to 
resist cross strains — such as a girder, or the tie-beam of a roof 
when supporting the ceiling. In such beams, the fibres of the 



190 AMERICAN IIOUSE-CARPENTER. 

wood in the upper part are compressed ; and therefore a simple butt 
joint at that place, (as in Fig. 240,) is far preferable to any other. 
In such case, an oblique joint is the very worst. The under 
side of the beam being in a state of tension, it must be indented 
or bolted, or both ; and an iron plate under the heads of the bolts, 
gives a great addition of strength. 

Scarfing requires accuracy and care, as all the indents should 
bear equally ; otherwise, one being strained more than another, 
there would be a tendency to splinter off the parts. Hence the 
simplest form that will attain the object, is by far the best. In all 
beams that are compressed endwise, abutting joints, formed at 
right angles to the direction of their length, are at once the simplest 
and the best. For a temporary purpose, Fig. 236 would do very 
well ; it would be improved, however, by having a piece bolted 
on all four sides. Fig. 237, and indeed each of the others, since 
they have no oblique joints, would resist compression v^ell. 

337.— In framing one beam into another for bearing pui poses, 
such as a floor-beam into a trimmer, the best place to make the 
mortice in the trimmer, is in the neutral line, (see Art. 254,) 
which is in the middle of its depth. Some have thought that, 
as the fibres of the upper edge are compressed, a mortice might 
be made there, and the tenon be driven in tight enough to make 
the parts as capable of resisting the compression, as they would 
be without it ; and they have therefore concluded that plan to be 
the best. This could not be the case, even if the tenon would 
not shrink ; for a joint between two pieces cannot possibly be 
made to resist compression, so well as a solid piece without joints. 
The proper place, therefore, for the mortice, is at the middle of 
the depth of the beam ; but the best place for the tenon, in the 
floor-beam, is at its bottom edge. For the nearer this is placed to 
the upper edge, the greater is the liability for it to splinter off; if 
the joint is formed, therefore, as at Fig. 241, it will combine all 
the advantages that can be obtained. Double tenons are objec- 
tionable, because the piece framed into is needlessly weakened, 



FRAMING. 



191 



1 



t'lg. '241. 



and the tenons are seldom so accurately made as to bear equally. 
For this reason, unless the tusk at a in the figure fits exactly, so 
as to bear equally with the tenon, it had better be omitted. And 
in sawing the shoulders, care should be taken not to saw into the 
tenon in the least, as it would wound the beam in the place least 
able to bear it. 

338. — Thus it will be seen that framing weakens both pieces, 
more or less. It should, therefore, be avoided as much as possi- 
ble ; and where it is practicable one piece should rest njw?i the 
other, rather than be framed into it. This remark applies to the 
bridging-joists in a framed floor, to the purlins and jack-rafters of 
a roof, tfec. 




Fig. 242 



339. — In a framed truss for a roof, bridge, partition, &c., the 
joints should be so constructed as to direct the pressures through 
the axes of the several pieces, and also to avoid every tendency 
of the parts to slide. To attain this object, the abutting surface 
on the end of a strut should be at right angles to the direction of 
the pressure ; as at the joint shown in Fig. 242 for the foot of a 
rafter, (see Art. 257,) in Fig. 243 for the head of a rafter, and in 
Fig. 244 for the foot of a strut or brace. The joint at Fig. 242 
\s not cut completely across the tie-beam, but a narrow lip is left 



192 



AMERICAN HOUSE-CARPENTER. 



Standing in the middle, and a corresponding indent is made in 
the rafter, to prevent the parts from separating sideways. The 
abutting surface should be made as large as the attainment of 
other necessary objects will admit. The iron strap is added to 
prevent the rafter from sliding out, should the end of the tie-beamj 
by decay or otherwise, splinter off. In making the joint shown 
at Fig. 243, it should be left a little open at a, so as to bring the 
parts to a fair bearing at the settling of the truss, which must 
necessarily take place from the shrinking of the king-post and 
other parts. If the joint is made fair at first, when the truss 
settles it will cause it to open at the under side of the rafter, thus 
throwing the whole pressure upon the sharp edge at a. This will 
cause an indentation in the king-post, by which the truss will be 
made to settle further ; and this pressure not being in the axis of 
the rafter, it will be greatly increased, thereby rendering the rafter 
liable to split and break. 




h 



Fiff. 245. 



Fig. 246. 



c:^^' 



Fig. 247. 



340. — If the rafters and struts were made to abut end to end, 
as in Fig. 245, 246 and 247, and the king or queen post notched 
on in halves and bolted, the ill effects of shrinking would be 
avoided. This method has been practised with success, in some 
of the most celebrated bridges and roofs in Europe ; and, were 
its use adopted in this country, the unseemly sight of a hogged 
ridge would seldom be met with. A plate of cast iron between 
the abutting surfaces, will equdize the pressure. 



FRAMING. 193 





Fig. 248. Fig. 249. 



341. — Fig. 248 is a proper joint for a collar-beam in a small 
roof: the principle shown here should characterize all tie-joints. 
The dovetail joint, although extensively practised in the above 
and similar cases, is the very worst that can be employed. The 
shrinking of the timber, if only to a small degree, permits the tie 
to withdraw — as is shown at Fig. 249. The dotted line shows 
the position of the tie after it has shrunk. 

342. — Locust and white-oak pins are great additions to the 
strength of a joint. In many cases, they would supply the place 
of iron bolts ; and, on account of their small cost, they should be 
used in preference wherever the strength of iron is not requisite. 
In small framing, good cut nails are of great service at the joints ; 
but they should not be trusted to bear any considerable pressure, 
as they are apt to be brittle. Iron straps are seldom necessary, as all 
the joinings in carpentry may be made without them. They can 
be used to advantage, however, at the foot of suspending-pieces, 
and for the rafter at the end of the tie-beam. In roofs for ordi- 
nary purposes, the iron straps for suspending-pieces may be as 
follows : When the longest unsupported part of the tie-beam is 
10 feet, the strap may be 1 inch wide by j\ thick. 
15 " " U " i " 

20 " " 2 " i " 

In fastening a strap, its hold on the suspending-piece will be much 
increased, by turning its ends into the wood. Iron straps should 
be protected from rust ; for thin plates of iron decay very soon 

26 



IQ'* AMERICAN HOUSE-CARPENTER. 

especially when exposed to dampness. For this purpose, as soon 
as the strap is made, let it be heated to about a blue heat, and, 
while it is hot, pour over its entire surface raw linseed oil, or rub 
it with beeswax. Either of these will give it a coating whici' 
dampness will not penetiat>3. 



SECTION v.— DOORS, WINDOWS, &^. 



DOORS. 

343. A.mong the several architectural arrangements of an edi- 
fice, the door is by no means the least in importance ; and, if pro- 
perly constructed, it is not only an article of use, but also of or- 
nament, adding materially to the regularity and elegance of the 
apartments. The dimensions and style of finish of a door, should 
be in accordance with the size and style of the building, or the 
apartment for which it is designed. As regards the utility of 
doors, the principal door to a public building should be of suflii- 
cient width to admit of a free passage for a crowd of people ; 
while that of a private apartment will be wide enough, if it per- 
mit one person to pass without being incommoded. Experience 
has determined that the least width allowable for this is 2 feet 8 
inches ; although doors leading to inferior and unimportant rooms 
may, if circumstances require it, be as narrow as 2 feet 6 inches ; 
and doors for closets, where an entrance is seldom required, may 
be but 2 feet wide. The width of the principal door to a public 
building may be from 6 to 12 feet, according to the size of the 
building ; and the width of doors for a dwelling may be from 2 
feet 8 inches, to 3 feet 6 inches. If the importance of an apart- 
ment in a dwelling be such as to require a door of greater width 



196 AMERICAN HOUSE-CARPENTER. 

than 3 feet 6 inches, the opening should be closed with two 
doors, or a door in two folds ; generally, in such cases, where the 
opening is from 5 to 8 feet, folding or sliding doors are adopted. 
As to the height of a door, it should in no case be less than about 
6 feet 3 inches ; and generally not less than 6 feet 8 inches. 

344. — The proportion between the width and height of single 
doors, for a dwelling, should be as 2 is to 5 ; and, for entrance- 
doors to public buildings, as 1 is to 2. If the width is given and 
the height required of a door for a dwelling, multiply the width 
by 5, and divide ttie product by 2 ; but, if the height is given and 
the width required, divide by 5, and multiply by 2. Where two 
or more doors of different widths show in the same room, it is 
well to proportion the dimensions of the more important by the 
above rule, and make the narrower doors of the same height as 
the wider ones ; as all the doors in a suit of apartments, except 
the folding or sliding doors, have the best appearance when of 
one height. The proportions for folding or sliding doors should 
be such that the width may be equal to | of the height ; yet this 
rule needs some qualification : for, if the width of the opening 
be greater than one-half the width of the room, there will not be 
a sufficient space left for opening the doors ; also, the height 
should be about one-tenth greater than that of the adjacent single 
doors, 

345. — Where doors have but two panels in width, let the stiles 
and muntins be each | of the width ; or, whatever number of 
panels there may be, let the united widths of the stiles and the 
muntins, or the whole width of the solid, be equal to | of the width 
of the door. Thus : in a door, 35 inches wide, containing two 
panels in width, the stiles should be 5 inches wide ; and in a door, 
3 feet 6 inches wide, the stiles should be 6 inches. If a door, 3 
feet 6 inches wide, is to have 3 panels in width, the stiles and 
muntins should be each 4^ inches wide, each panel being 8 inches. 
The bottom rail and the lock rail ought to be each equal in 
width to tV of the height of the door ; and the top rail, and all 



DOORS, WINDOWS, &C. 



197 



others, of the same width as the stiles. The moulding on the 
panel should be equal in width to i of the width of the stile. 




Fig. 250. 



346. — Fig. 250 shows an approved method of trimming doors : 
a is the door stud ; 6, the lath and plaster ; c, the ground ; d, tfie 
jamb ; e, the stop ; /and g^ architrave casings ; and A, the door 
stile. It is customary in ordinary work to form the stop for the 
door by rebating the jamb. But, when the door is thick and 
heavy, a better plan is to nail on a piece as at e in the figure. 
This piece can be fitted to the door, and put on after the door is 
hung ; so, should the door be a trifle winding, this will correct 
the evil, and the door be made to shut solid. 

347. — Fig. 251 is an elevation of a door and trimmings suita- 
ble for the best rooms of a dwelling. (For trimmings generally, 
see Sect. III.) The number of panels into which a door should 
be divided, is adjusted at pleasure 5 yet the present style of finish- 
ing requires, that the number be as small as a proper regard for 
strength will admit. In some of our best dwellings, doors have 
been made having only two upright panels. A few years expe- 
rience, however, has proved that the omission of the lock rail 
is at the expense of the strength and durability of the door ; 9 
four-panel door, therefore, is the best that can be made. 

348. — The doors of a dwelling should all be hung so as to open 
into the principal rooms ; and, in general, no door should be hung 
to open into the hall, or passage. As to the proper edge of the 
door on which to aflix the hinges, no general rule can be assigned 



198 



AMERICAN HOUSE-CARPENTER. 



aK 



liliaiLIfLMl^tl^mmtJi^JiiAiiiiiiyfnffmiiiJi^j^uMiti mM^um i Lf.mmMJL i LJtnii 



C 



_ 



Fig. 251. 



It may be observed, however, that a bed-room door should be 
hung so that, when half open, it will screen the bed ; and a door 
leading from a hall, or passage, to a principal room, should screen 
the fire. 



WINDOWS. 

349. — A window should be of such dimensions, and in such a 
position, as to admit a sufficiency of light to that part of the 
apartment for which it is designed. No definite rule for the size 



DOORS, WINDOWS, &C. l99 

can well be given, that will answer in all cases ; yet, as an ap- 
proximation, the following has been used for general purposes. 
Multiply together the length and the breadth in feet of the apart- 
ment to be lighted, and the product by the height in feet ; then 
the square-root of this product will show the required number of 
square feet of glass. 

350. — To ascertain the dimensions of window frames, add 4J 
inches to the width of the glass for their width, and 6^ inches to 
the height of the glass for their height. These give the dimen- 
sions, in the clear, of ordinaiy frames for 12-light windows ; the 
height being taken at the inside edge of the sill. In a brick wall, 
the width of the opening is 8 inches more than the width of the 
glass — 4^ for the stiles of the sash, and 3^ for hanging stiles — 
and the height between the stone sill and lintel is about 10 g inches 
more than the height of the glass, it being varied according to the 
thickness of the sill of the frame. 

351. — In hanging inside shutters to fold into boxes, it is ne- 
cessary to have the box shutter about one inch wider than the 
flap, in order that the flap may not interfere when both are folded 
into the box. The usual margin shown between the face of the 
shutter when folded into the box and the quirk of the stop bead, 
or edge of the casing, is half an inch ; and, in the usual method 
of letting the whole of the thickness of the butt hinge into the 
edge of the box shutter, it is necessary to make allowance for the 
throw of the hinge. This may, in general, be estimated at i of 
an inch at each hinging ; which being added to the margin, the 
entire width of the shutters will be 1^ inches more than the width 
of the frame in the clear. Then, to ascertain the width of the 
box shutter, add 1^ inches to the width of the frame in the clear, 
between the pulley stiles ; divide this product by 4, and add 
half an inch to the quotient ; and the last product will be the re- 
quired width. For example, suppose the window to have 3 
lights in width, 11 inches each. Then, 3 times 11 is 33, and 4^ 
added for the wood of the sash, gives 37^ 2>7i and 1^ is 39 



200 AMERICAN HOUSE CARPENTER. 

and 39; divided by 4, gives 9| ; to which add half an inch, and 
the result will be 10^: inches, the width required for the box shutter. 
352. — In disposing and proportioning windows for the walls of 
a building, the rules of architectural taste require that they be of 
different heights in different stories, but of the same width. The 
windows of the upper stories should all range perpendicularly 
over those of the first, or principal, story; and they should be 
disposed so as to exhibit a balance of parts throughout the front 
of the building. To aid in this, it is always proper to place the 
front door in the middle of the front of the building ; and, where 
the size of the house will admit of it, this plan should be adopted. 
(See the latter part of Art. 214.) The proportion that the height 
should bear to the width, may be, in accordance with general 
usage, as follows : 

The height of basement windows, 1^ of the width. 
" " principal-story " 2| " 

" " second-story " If " 

« " third-story " 1| « 

" " fourth-story " If " 

" " attic-story " the same as the width. 

But, in determining the height of the windows for the several 
stories, it is necessary to take into consideration the height of the 
story in which the window is to be placed. For, in addition to 
the height from the floor, which is generally required to be from 
28 to 30 inches, room is wanted above the head of the window 
for the window-trimming and the cornice of the room, besides 
some respectable space which there ought to be between these. 

353. — The present style of finish requires the heads of win- 
dows in general to be horizontal, or square-headed ; yet, it is well 
to be possessed of information for trimming circular-headed win- 
dows, as repairs of these are occasionally needed. If the jambs 
of a door or window be placed at right angles to the face of the 
wall, the edges of the sojfif, or surface of the head, would be 
straight, and its length be found by getting the stretch-out of the 



DOORS, WINDOWS, &C. 



201 



circle, {Art. 92 ;)but, when the jambs are placed obliquely to the 
face of the wall, occasioned by the demand for light in an 
oblique direction, the form of the soffit will be obtained as in the 
ollowing article : and, when the face of the wall is circular, as in 
the succeeding one. 

/ 




Fig. 252. 



354. — To find the for ?n of the soffit for circular window- 
heads^ when the light is received in an oblique direction. Let 
a b cdj {Fig. 252,) be the ground-plan of a given window, ande/ 
a, a vertical section taken at right angles to the face of the jambs. 
From a, through e, draw ag, at right angles to a b ; obtain the 
stretch-out of efa^ and make e g equal to it ; divide e g and e 
f a, each into a like number of equal parts, and drop perpen- 
diculars from the points of division in each ; from the points of 
intersection, 1, 2, 3, &c., in the line, a d, draw horizontal lines to 
meet corresponding perpendiculars from eg; then those points 
of intersection will give the curve line, d g, which will be the 
one required for the edge of the soffit. The other edge, c A, is 
found in the same manner. 

355. — To find the form of the soffit for circular window- 
heads^ when the face of the wall is curved. Let abed, {Fig. 
253,) be the ground-plan of a given window, and efa.B. vertical 
section of the head taken at right angles to the face of the jambs. 

26 



202 



AMERICAN HOUSE-CARPENTER. 




Fi',^ x:53. 



Proceed as in the foregoing article to obtain the line, d g ; then 
that will be the curve required for the edge of the soffit ; the 
other edge being found in the same manner. 

If the given vertical section be taken in a line with the face of 
the wall, instead of at right angles to the face of the jambs, place 
it upon the line, c 6, [Fig. 252 ;) and, having drawn ordinates at 
right angles to c 6, transfer them to ef a ; in this way, a section 
at right angles to the jambs can be obtained. 



SECTION VL— STAIRS. 



356. — The stairs is that m»3chtinlciil arrangement in a build- 
ing by which access is obtained from one story to another. Their 
position, form and finish, when determined with discriminating 
taste, add greatly to the comfort and elegance of a structure. As 
regards their position, the first object should be to have them near 
the middle of the building, in order that an equally easy access 
may be obtained from all the rooms and passages. Next in im- 
portance is light; to obtain which they would seem to be best 
situated near an outer wall, in which windows might be construc- 
ted for the purpose ; yet a sky-light, or opening in the roof, would 
not only provide light, and so secure a central position for the 
stairs, but may be made, also, to assist materially as an ornament 
to the building, and, what is of more importance, afford an op- 
portunity for better ventilation. 

357. — It would seem that the length of the raking side of the 
pitch-board, or the distance from the top of one riser to the top of 
the next, should be about the same in all cases ; for, whether stairs 
be intended for large buildings or for small, for public or for pri- 
vate, the accommodation of men of the same stature is to be con- 
sulted in every instance. But it is evident that, with the same 
effort, a longer step can be taken on level than on rising ground ; 



204 



AMERICAN HOUSE-CARPENTER. 



and that, although the tread and rise cannot be proportioned 
merely in accordance with the style and importance of the build- 
ing, yet this may be done according to the angle at which the 
flight rises. If it is required to ascend gradually and easy, the 
length from the top of one rise to that of another, or the hypothe 
nuse of the pitch-board, may be long ; but, if the flight is steep, 
the length must be shorter. Upon this data the followiug problem 
is constructed. 




3.58. — To proportion the rise and tread to one another. 
Make the line, a b, {Fig. 254,) equal to 24 inches ; from b, erect 
b c, at right angles to a 6, and make b c equal to 12 inches ; join a 
and c, and the triangle, a b c, will form a scale upon which to 
graduate the sides of the pitch-board. For example, suppose a 
very easy stairs is required, and the tread is fixed at 14 inches. 
Place it from b to/, and from/; draw f g^ at right angles to a b ; 
then the length oi f g will be found to be .5 inches, which is a 
proper rise for 14 inches tread, and the angle, f b g^ will show 
the degree of inclination at which the flight will ascend. But, in 
a majority of instances, the height of a story is fixed, while the 
length of tread, or the space that the stairs occupy on the lower 
floor, is optional. The height of a story being determined, the 
height of each rise will of course depend upon the number into 
which the whole height is divided ; the angle of ascent being more 
easy if the number be great, than if it be smaller. By dividing 



STAIRS. 205 

the whole height of a story into a certain number of rises, sup- 
pose the length of each is found to be 6 inches. Place this length 
from h to A, and draw h i, parallel to a b ; then h i, or b j will be 
the proper tread for that rise, and J b i will show the angle of as- 
cent. On the other hand, if the angle of ascent be given, as a 
b l^ {b I being lOJ inches, the proper length of run for a step- 
ladder,) drop the perpendicular, I /;, from I Xo k ; then I kb will 
be the proper proportion for the sides of a pitch-board for that 
run. 

359. — The angle of ascent will vary according to circum- 
stances. The following treads will determine about the right in- 
clination for the different classes of buildings specified. 

In public edifices, tread about 14 inches. 

In first-class dwellings " 12^ " 

In second-class " "11 " 

In third-class " and cottages " 9 " 

Step-ladders to ascend to scuttles, (fee, should have from 10 to 
11 inches run on the rake of the string. (See notes at Ai^t. 103.) 
360. — The length of the steps is regulated according to the ex- 
tent and importance of the building in which they are placed, 
varying from 3 to 12 feet, and sometimes longer. Where two per- 
sons are expected to pass each other conveniently, the shortest 
length that will admit of it is 3 feet ; still, in crowded cities where 
land is so valuable, the space allowed for passages being very 
small, they are frequently executed at 2 J feet. 

361. — To find the dimensions of the pitch-board. The first 
thing in commencing to build a stairs, is to make the /^zVcA-board ; 
this is done in the following manner. Obtain very accurately, in 
feet and inches, the perpendicular height of the story in which 
the stairs are to be placed. This must be taken from the top ot 
the floor in the lower stoiy to the top of the floor in the upper 
story. Then, to obtain the number of rises, the height in inches 
thus obtained must be divided by 5, 6, 7, 8, or 9, according to the 
quality and style of the building in which the stairs are to be 



206 AMERICAN HOUSE-CARPENTER. 

built. For instance, suppose the building to be a first-class 
dwelling, and the height ascertained is 13 feet 4 inches, or 160 
inches. The proper rise for a stairs in a house of this class is 
about 6 inches. Then, 160 divided by 6, gives 26f inches. This 
being nearer 27 than 26, the number of risers, should be 27. 
Then divide the height, 160 inches, by 27, and the quotient will 
give the height of one rise. On performing this operation, the 
quotient will be found to be 5 inches, | and ~ of an inch. 

Then, if the space for the extension of the stairs is not limited, 
the tread can be found as at Art. 358. But, if the contrary is the 
case, the whole distance given for the treads must be divided by 
the number of treads required. On account of the upper floor 
forming a step for the last riser, the number of treads is always 
one less than the number of risers. Having obtained this 
rise and tread, the pitch-board may be made in the follow- 
ing manner. Upon a piece of well-seasoned board about | of an 
inch thick, having one edge jointed straight and square, lay the 
corner of a carpenters'-square, as shown at Fig. 255. Make a h 




Fig. 255. 

equal to the rise, and h c equal to the tread ; mark along those 
edges with a knife, and cut it out by the marks, making the edges 
perfectly square. The grain of the wood must run in the direction 
indicated in the figure, because, if it shrinks a trifle, the rise and 
the tread will be equally affected by it. When a pitch-board is 
first made, the dimensions of the rise and tread should be pre- 
served in figures, in order that, should the first shrink, a second 
could be made. 

362. — To lay out the string. The space required for timber 




2ivr 



Fitr. 25(j. 



and plastering under the steps, is about 5 inches for ordinary 
stairs ; set a gauge, therefore, at 5 inches, and run it on the lower 
edge of the plank, as a 6, {Fig. 256.) Connnencing at one end, 
lay the longest side of the pitch-board against the gauge-mark, a 
6, as at c, and draw by the edges the lines for the first rise and 
tread; then place it successively as at d, e and/, until the re- 
quired number of risers shall be laid down. 




F7 



UJ. 



Fig. 257. 



363. — Fig. 257 represents a section of a step and riser, joined 
after the most approved method. In this, a represents the end of 
a block about 2 inches long, two of which are glued in the corner 
in the length of the step. The cove at b is planed up square, 
glued in, and stuck after the glue is set. 



PLATFORM STAIRS. 

364. — A platform stairs ascends from one story to another in 
two or more flights, having platforms between for resting and 
to cnange their direction. This kind of stairs is the most easily 
constructed, and is therefore the most common. The cyliu 



20n 



AMERICAN HOUSE-CAIIPENTBE. 




Fig. 258. 



der is generally of small diameter, in most cases about 6 inches. 
It may be worked out of one solid piece, but a better way is to 
glue together three pieces, as in Fig. 258; in which the piece.^j 
a, b and c, compose the cylinder, and d and e represent parts of 
the strings. The strings, after being glued to the cylinder, are 
secured with screws. The joining at o and r? is the most proper 
for that kind of joint. 

36.5. — 7^0 obtain the form of the lower edge of the cylinder. 
Find the stretch-out, d e, {Fig. 259,) of the face of the cylinder, 
a 6 c, according to Art. 92 ; from d and e, draw d f and e g, at 
right angles to d e ; draw h g, parallel to d e, and make hf and 
g i, each equal to one rise; from i and/, draw ij and/ A:, paral- 
lel to h g ; place the tread of the pitch-board at these last lines, 
and draw by the lower edge the lines, k h and i I ; parallel to 
tnese, draw m n and o p, at the requisite distance for the dimen- 
sions of the string : from 5, the centre of the plan, draw s q. 
parallel to df; divide h qand q g, each into 2 equal parts, as at 
V and w; from v and w, draw v n and w o, parallel to/ c?; join n 
and 0, cutting q s m r ; then the angles, u n r and rot, being 
eased off according to Art. 89, will give the proper curve for the 
bottom edge of the cylinder. A centre may be found upon which 
to describe these curves thus : from w, draw u :r, at right angles 
to mn; from r, draw r a:, at right angles to no ; then x will be 
the centre for the curve, u r. The centre for the curve, r ^, is 
found in the same manner. 



STAIRS. 



209 




Fig. 259. 



366. — To find the position for the balusters. Place the 
centre of the first baluster, (6. Fig. 260,) 5 its diameter from the 
face of the riser, c c?, and i its diameter from the end of the step, 
e d ; and place the centre of the other baluster, a, half the tread 
from the centre of the first. The centre of the rail must be placed 
over the centre of the balusters. Their usual length is 2 feet 
5 inches, and 2 feet 9 inches, for the short and the long balusters 
respectively. 




a- 



^ 



Piir sm. 



27 



210 



AMERICAN HOUSE-CARPENTER. 




Fig. 261. 



367. — To find the face-mould for a round hand-rail to plat- 
form stairs. Case 1. — When the cylinder is small. In Fig. 
261, J and e represent a vertical section of the last two steps of the 
first flight, and d and i the first two steps of the second flight, of 
a platform stairs, the line, e /, being the platform ; and a 6 c is 
the plan of a line passing through the centre of the rail around 
the cylinder. Through i and d, draw i k, and through J and e, 
draw 7 k ; from A;, draw k Z, parallel to /e ; from 6, draw h m, 
parallel tog d; from /, draw I r, parallel to kj ; from n, draw n 
/, at right angles to jf k : on the line, o b. maki. o t equal to w ^ ; 
join c and ^ : on ie line, j c, {Fig. 262,) make e c equal to e w at 
Fig. 261 ; from c, draw c t^ ai nt'ht angles Xoj c, and make c t 



STAIRS. 21J 

i I 




Fig. 262. 



equal to c ^ at Fig. 261 ; through t, drawp /, parallel ioj c, and 
make 1 1 equal to ^ / at Fig. 261 : join I and c, and complete the 
parallelogram, eels; find the points, o, o, o, according to ^r<f. 
118 ; upon e, o, o, o, and/, successively, with a radius equal to 
half the width of the rail, describe the circles shown in the figure ; 
then a curve traced on both sides of these circles and just touch- 
ing them, will give the proper form for the mould. The joint at 
I is drawn at right angles to c I. 

368. — Elucidation of the foregoing method. This excellent 
plan for obtaining the face-moulds for the hand-rail of a platform 
stairs, has never before been pnblished. It was communicated to 
me by an eminent stair-builder of this city : and having seen 
rails put up from it, I am enabled to give it my unqualified re- 
commendation. In order to have it fully understood, I have in- 
troduced Fig. 263 ; in which the cylinder, for this purpose, i? 
made rectangular instead of circular. The figure gives a per- 
spective view of a part of the upper and of the lower flights, and 
a part of the platform about the cylinder. The heavy lines, i wi, 
m c and cj, show the direction of the rail, and are supposed to 
pass through the centre of it. When the rake of the second 
flight is the same as that of the first, which is here and is gene- 
rally the case, the face-mould for the lower twist will, when re- 
versed, do for the upper flight: that part of the rail, therefore, 
which passes from e to c and from c to Z, is all that will need ex- 
planation. 

Suppose, then, that the parallelogram, e a o c, represent a plane 
lying perpendicularly over e ah f being inclined in the direction, 
e c, and level in the direction, c o ; suppose this plane, e a o c, 



212 



AMERICAN HOUSE-CARPENTER. 




Fi?. 263. 



be revolved on e c as an axis, in the manner indicated by the arcs, 
o n and a x^ until it coincides with the plane, e r t c ; the line, a 
0, will then be represented by the line, x n ; then add the paral- 
lelogram, xrt n^ and the triangle, ctl^ deducting the triangle, ers; 
and the edges of the plane, e s I c. inclined in the direction, ec, and 
also in the direction, c I, will lie perpendicularly over the plane, e 
a bf. From this we gather that the line, c o, being at right angles to 



STAIRS. 



213 



e c, must, in order to reach the point, Z, be lengthened the distance, 
n t^ and the right angle, e c ^, be made obtuse by the addition to 
it of the angle, t c I. By reference to Fig. 261, it will be seen 
that this lengthening is performed by forming the right-angled 
triangle, c o t^ corresponding to the triangle, c o ^, in Fig, 263. 
The line, c Z, is then transferred to Fig. 262, and placed at right 
angles to e c ; this angle, e c t^ being increased by adding the an- 
gle, t c Ij corresponding to ^ c /, Fig. 263, the point, Z, is reached, 
and the proper position and length of the lines, e c and c I ob- 
tained. To obtain the face-mould for a rail over a cylindrical 
well-hole, the same process is necessary to be followed until the 
the length and position of these lines are found ; then, by forming 
the parallelogram, eels, and describing a quarter of an ellipse 
therein, the proper form will be given. 




Fig 264. 



369. — Case 2. — When the cylinder is large. Fisr. 264 re- 



214 



AMERICAN HOUSE-CARPENTER. 



presents a plan and a vertical section of a line passing through the 
centre of the rail as before. From 6, draw h k, parallel toed; ex- 
tend the lines, i d and J e, until they meet khiuk and/; from n, 
draw n I, parallel to o 6 ; through I, draw 1 1, parallel to j k ; from 
k, draw k t, at right angles to j k ; on the line, o 6, make o t equal 
to A; t. Make e c, (Fi^. 265,) equal to e A: at Fig. 264 ; from c, 




Fisr. 263. 



draw c ^, at right angles to e c, and equal to c ^ at Fig, 264 ; from 
t, draw ^ ^, parallel to c e, and make 1 1 equal to ^ Z at Fig. 264 ; 
complete the parallelogram, eels, and find the points, o, o, o, as 
before : then describe the circles and complete the mould as in 
Fio\ 262. The difference between this and Case 1 is, that the 
line, c t, instead of being raised and thrown out, is lov/ered and 
drawn in. (See note at y^^^. 2.55.^ 




Fig. -^eu c 

370._Oase 3. — Where the rake meets the level. In Fig. 



STAIRS. 215 

266, a b cis the plan of a line passing through the centre of the 
rail around the cylinder as before, and J and e is a vertical section 
of two steps starting from the floor, h g. Bisect e hin d, and 
through d, draw df, parallel to h g ; bisect/ n in Z, and from /, 
draw / t, parallel to nj; from w, draw n t, at right angles to / n ; 
on the line, o 6, make o t equal to 71 1. Then, to obtain a mould 
for the twist going up the flight, proceed as at Fig. 262 ; making 
c cin that figure equal to e n in Fig, 266, and the other lines ot 
a length and position such as is indicated by the letters of reference 
in each figure. To obtain the mould for the level rail, extend 6 
0, {Fig. 266,) to i ; make i equal to/ Z, and join i and c ; make 
c i, {Fig. 267,) equal to c i at Fig. 266 ; through c, draw c c?, at 




d 

Fig. 267. 



right angles to c i ; make d c equal to tZ/ at Fig. 266, and com- 
plete the parallelogram, odd; then proceed as in the previous 
cases to find the mould. 

371. — All the moulds obtained by the preceding examples have 
been for round rails. For these, the mould may be applied to 
a plank of the same thickness as the rail is intended to be, and 
the plank sawed square through, the joints being cut square from 
the face of the plank. A twist thus cut and truly rounded will 
hang in a proper position over the plan, and present a perfect and 
graceful wreath. 

372. — To bore for the balusters of a round rail before round- 
ing it. Make the angle, c t^ [Fig. 268,) equal to the angle, 
c Z, at Fig. 261 ; upon c, describe a circle with a radius equal to 
half the thickness of the rail ; draw the tangent, b rf, parallel to 
t c, and complete the rectangle, e b df having sides tangical to 
the circle ; from c, draw c a, at right angles to oc ; then, b d 
being the bottom of the rail, set a gauge from b to a, and run it 
the whole length of the stufl*; in boring, place the centre of the 



216 



AM ER iCA \ [ T O U P I:>CA RPENTER. 

h 

d 




i- lu. -I'S. 



bit in the gauge-mark at a. and bore in the direction, a c. To do 
this easily, make chucks as represented in the figure, the bottom 
edge, g- h, being parallel to o c, and having a place sawed out, as 
ef, to receive the rail. These being nailed to the bench, the rail 
will be held steadily in its proper place for boring vertically. 
The distance apart that the balusters require to be, on the under 
side of the rail, is one-half the length of the rake-side of the 
pitch-board. 




Fig. 269 



STAIRS. 



217 



373. — To obtain, by the foregoing principles, the face-mould 
for the twists of a moulded rail upon platform stairs. In Fig. 
269, a b c is the plan of a line passing through the centre of 
the rail around the cylinder as before, and the lines above 
it are a vertical section of steps, risers and platform, v\^ith 
the hnes for the rail obtained as in Fig. 261. Set half the width 
of the rail from b to f and from b to r, and from / and r, draw/ 
e and r d, parallel to c a At Fig. 270, the centre lines of the 



s 


d 


n e 


ml ^ 


/t~~ 


iHv 


\ Ic/ 


i 
c 


// 


{ 1 ^^ 


, 


g 


J 





Fig. 270. 

rail, k c and c n, are obtained as in the previous examples. Make 
c i and c j, each equal to c i at Fig. 269, and draw the lines, i m 
and j g, parallel to c k ; make n e and n d equal to n e and n d at 
Fig. 269, and draw d o and e I, parallel to n c ; also, through k, 
draw s g, parallel to n c ; then, in the parallelograms, m s d o and 
g s e I, find the elliptic curves, d m and e g, according to Art. 
118, and they will define the curves. The line, d e, being drawn 
through n parallel to k c, defines the joint, which is to be cut 
through the plank vertically. If the rail crosses the platform rather 
steep, a butt joint will be preferable, to obtain which see Art. 405. 




218 AMERICAN HOUSE-CARPENTEJl. 

374. — To apply the mould to the plank. The mould obtained 
according to the last article must be applied to both sides of the 
plank, as shown at Fig. 271. Before applying the mould, the 
edge, e/, must be bevilled according to the angle, c t x^dX Fig. 
269 ; if the rail is to be canted up, the edge must be bevilled at 
an obtuse angle with the upper face ; but if it is to be canted 
down, the angle that the edge makes with the upper face mnst be 
acute. From the spring of the curve, a, and the end, c, draw 
vertical lines across the edge of the plank by applying the pitch- 
board, a b c ; then, in applying the mould to the other side, place 
the points, a and c, at b and/; and, after marking around it, saw 
the rail out vertically. After the rail is sawed out, the bottom 
and the top surfaces must be squared from the sides. 

375. — To ascertain the thickness of stuff required for the 
twists. The thickness of stuff required for the twists of a round 
rail, as before observed, is the same as that for the straight ; but 
for a moulded rail, the stuff for the twists must be thicker than 
that for the straight. In Fig. 269, draw a section of the rail be- 
tween the lines, d r and e f and as close to the line, d e, as possi- 
ble ; at the lower corner of the section, draw g A, parallel to d e; 
then the distance that these lines are apart, will be the thickness 
required for the twists of a moulded rail. . 

The foregoing method of finding moulds for rails is applicable 
to all stairs which have continued rails around cylinders, and are 
without winders. 

WINDING STAIRS. 

376. — Winding stairs have steps tapering narrower at one end 
than at the other. In some stairs, there are steps of parallel width 
incorporated with tapering steps ; the former are then called ^yer* 
and the latter tiinders. 

377. — To describe a regular geometrical winding stairs. 
In Fig. 272, abed represents the inner surface of the wall en- 
closing the space allotted to the stairs, a e the length of the steps, 
and efgh the cylinder, or face of the front string. The line, 



STAIRS. 



219 




Fig. 272. 



a e, is given as the face of the first riser, and the point, j^ for the 
limit of the last. Make e i equal to 18 inches, and upon o, with 
i for radius, describe the arc, ij; obtain the number of risers 
and of treads required to ascend to the floor at j, according to Art. 
361, and divide the arc, ij, into the same number of equal parts 
as there are to be treads ; through the points of division, 1, 2, 3. 
&c., and from the wall-string to the front-string, draw lines tend- 
ing to the centre, o ; then these lines will represent the face of 
each riser, and determine the form and width of the steps. Allow 
the necessary projection for the nosing beyond a e, which should 
be equal to the thickness of the step, and then a el k will be the 
dimensions for each step. Make a pitch-board for the wall-string 
having a k for the tread, and the rise as previously ascertained ; 
with this, lay out on a thicknessed plank the several risers and 
treads, as at Fig, 256, gauging from the upper edge of the string 
for the line at which to set the pitch-board. 

Upon the back of the string, with a 1;^ inch dado plane, make 



220 AMERICAN HOUSE-CARPENTER. 

a succession of grooves IJ inches apart, and parallel with the 
lines for the risers on the face. These grooves must be cut along 
the whole length of the plank, and deep enough to admit of the 
plank's bending around the curve, abed. Then construct a 
drum, or cylinder, of any common kind of stuff, and made to fit 
a curve having a radius the thickness of the string less than o a ; 
upon this the string must be bent, and the grooves filled with strips 
of wood, called ke^s, which must be very nicely fitted and glued 
in. After it has dried, a board thin enough to bend around on the 
outside of the string, must be glued on from one end to the other, 
and nailed with clout nails. In doing this, be careful not to nail 
into any place where a riser or step is to enter on the face. 

After the string has been on the drum a sufficient time for the 
glue to set, take it off, and cut the mortices for the steps and 
risers on the face at the lines previously made ; which may be 
done by boring with a centre-bit half through the string, and 
nicely chiseling to the line. The drum need not be made so 
large as the whole space occupied by the stairs, but merely large 
enough to receive one piece of the wall-string at once — for it 
is evident that more than one will be required. The front string 
may be constructed in the same manner ; taking e I instead of a 
k for the tread of the pitch-board, dadoing it with a smaller dado 
plane, and bending it on a drum of the proper size. 




Fig. 273. 

378. — To find the shape and position of the timbers neces- 
sary to support a winding stairs. The dotted lines in Fig. 
272 show the proper position of the timbers as regards the plan : 
the shape of each is obtained as follows. In Fig. 273, the line, 
1 a, is equal to a riser, less the thickness of the floor, and the 
lines, 2 w, 3 w, 4 0, 5 /? and 6 q, are each equal to one riser. The 



STAIRS. 2li\ 

line, a 2, is equal to am in Fig. 272, the line, m 3 to m ?i in that 
figure, &c. In drawing this figure, commence at a, and make 
the lines, a 1 and a 2, of the length above specified, and draw 
them at right angles to each other ; draw 2 m, at right angles to 
a 2, and m 3, at right angles to m 2, and make 2 m and m 3 of 
the lengths as above specified : and so proceed to the end. Then, 
through the points, 1, 2, 3, 4, 5 and 6, trace the line, 1 b ; upon 
the points, 1, 2, 3, 4, &c., with the size of the timber for radius, 
describe arcs as shown in the figure, and by these the lower line 
may be traced parallel to the upper. This will give the proper 
shape for the timber, a b, in Fig. 272 ; and that of the others may 
be found in the same manner. In ordinary cases, the shape of 
one face of the timber will be sufiicient, for a good workman 
can easily hew it to its proper level by that ; but where great 
accuracy is desirable, a pattern for the other side may be found 
in the same manner as for the first. 

379. — To find the falling-mould for the rail of a loinding 
stairs. In Fig. 274, a cb represents the plan of a rail around 
half the cylinder, A the cap of the newel, and 1, 2, 3, &c., the 
face of the risers in the order they ascend. Find the stretch-out, 
e f of a c b, according to Art. 92; from o, through the point of 
the mitre at the newel-cap, draw o s ; obtain on the tangent, e d, 
the position of the points, s and h'* as. at t and/^ ; from e tf^ and 
/, draw e a;, t u,/^ g'^ and /A, all at right angles Xo e d ; make e 
g equal to one rise and/- ^^ equal to 12, as this line is drawn 
from the 12th riser ; from g^ through g^.^ draw^ i; make g x equal 
to about three-fourths of a rise, (the top of the newel, x, should 
be 3^ feet from the floor ;) draw x w, at right angles to e x^ and 
ease off the angle at w ; at a distance equal to the thickness of 

• In the above, the references, a^, U^, &c., are introduced for the first time. During the 
time taken to refer to the figure, the memory of the /otto of these may pass from the mind, 
while that of the sound alone remains ; they may then be mistaken for a 2, 6 2, &c. This 
can be avoided in reading by giving them a sound corresponding to their meaning, which 
IS second a second h, &c. or a second, b second. 



222 



AMERICAN HOUSE-CARPENTER. 




r[ie rail, draw v w y, parallel to x ii i; from the centre of the plan, 
0, draw o I, at right angles to e d ; bisect h n in p, and through 
/>, at right angles to,^' i, draw a line for the joint; in tlie same 
manner, draw the joint at k ; then x y will be the falling-mould 
for that part of the rail which extends from 5 to 6 on the plan. 

380. — To find the face-mould for the railof axoinding-stairs. 
From the extremities of the joints in the falling-mould, as A', z 
and y, (Fig. 274,) draw k a^, z 6^ and y c?, at right angles Xo e d : 
make h ^ equal to / d. Then, to obtain the direction of \\v?. 
joint, c^ c^, or 6^ d\ proceed as at Fig. 275, at which the parts are 



STAIRS. 



223 




Fig. 275. 



shown at half their fill! size. A is the plan of the rail, and B is 
the falling-monid : in which k z is the direction of the butt-joint. 
From kj draw k b, parallel to I o, and k e, at right angles to k b ; 
from 6, draw b /, tending to the centre of the plan, and from/, draw 
/ e, parallel to b k ; from Z, through e, draw I ^, and from i, draw i 
d, parallel toef; join d and 6, and d b will be the proper direction 



AMERICAN HOUSE-CARPENTER. 



for the joint on the plan. The direction of the joint on the other 
side, a c, can be found by transferring the distances, x h and o rf, 
Xo X a and o c. (See Art. 384.) 




Fi-. 27G. 



Having obtained the direction of the joint, make s r d b, (Fig. 
276,) equal to 5 r db'^ in Fig. 27 A ; through r and d, draw t a ; 
through 6' and from d, draw t u and d e, at right angles to t a ; 
make t u and d e equal to t u and 6^ m, respectively, in Fig. 274 ; 
from w, through e, draw u ; through 6, from r, and from as many 
other points in the line, t a, as is thought necessary, as/, h and ;, 
draw the ordinates, r c,fg, h i,j k and a ; from w, c, ^, ^J k^ e 
and 0, draw the ordinates, ii 1, c 2, g 3, i A, k 5, e 6 and 7, at 
right angles to it, ; make ?i 1 equal to ^ 5, c 2 equal to r 2, ^ 3 
equal to/ 3, &c., and trace the curve, 1 7, through the points 
thus found ; find the curve, c e, in the same manner, by transfer- 
ring the distances between the line, t a, and the arc, r d ; join 1 
and c, also e and 7 ; then, I c e 7 will be the face-mould required 
for that part of the rail which is denoted by the letters, s r d* b"^, 
on the plan at Fig. 27 L 

To ascertain the mould for the next quarter, make acje, (i'V^. 



STAIRS. 



22k 




Fig. 277. 



277,) equal to a' c^j e^ at Fig. 274 ; at any convenient height on 
the h'ne, d i, in that figure, draw q i\ parallel to e d ; through c 
and J, {Fig. 277,) draw b d ; through a, and from 7, draw b k and 
; 0, at right angles to b d ; make b k an&j equal to ^ k and q 
i, respectively, in Fig. 27 A ; from k, through 0, draw kf; and 
proceed as in the last figure to obtain the face-mould, A. 

381. — To ascertain the requisite thickness of stuff. Case 
1. — When the falling-mould is straight. Make h and k m, 
{Fig. 277,) equal to i y at Fig. 274 ; draw h i and m n, parallel 
tob d ; through the corner farthest from k f as n or ^, draw n ^, 
parallel to kf ; then the distance between k f and n i will give 
the thickness required. 

382. — Case 2. — When the falling-mould is curved. In FHg. 
278, sr db'is, equal tosr d^Wm. Fig. 27 L Make a c equal to the 
stretch-out of the arc, s 6, according to Art. 92, and divide a c and 
s b. each into a like number of equal parts ; from a and c. and from 
each point of division in the line, ac, draw ak^el^ (fee, at right an- 
gles to a c ; make a A; equal to ^ w in Fig. 274, and c; equal to 6' wi 

29 



22t 



AMERICAN HOUSE-CARPENTER. 




in that figure, and complete the ral ling-mould, kj, every way equal 
to M m in Fig. 274 ; from the points of division in the arc, sb, draw 
lines radiating towards the centre of the circle, dividing the arc. 
r d, in the same proportion ass bis divided ; from d and b, draw 
dtanAb u, at right angles to a d, and from j and v, dmwj u and v 
w, at right angles toj c ; then xtuwwiWhea. vertical projection 
of the joint, d b. Supposing every radiating line across s r d b~ 
corresponding to the vertical lines across A: ^-to represent a joint, 
find their vertical projection, as at 1, 2, 3, 4, 5 and 6 ; through the 
corners of those parallelograms, trace the curve lines shown in the 
figure ; then 6 u will be a helinet, or vertical projection, oisrdb. 
To find the thickness of plank necessary to get out this part of 
the rail, draw the line, z t, touching the upper side of the helinet 
m two places : through the corner farthest projecting from that 
line, as w, draw y w, parallel to ^ ^- then the distance between 
those Imes will be the proper thickness of stuff for this part of the 
rail. The same process is necessary to find the thickness of 
stuff in all cases in which the falling-mould is in any way curved. 
i»i— To apply the face-mould to the plank. In Pig 279 
A represents the plank with its best side and edge in view and 
B the same plank turned up so as to bring in view the other side 



STAIRS. 



227 




Fig. 279. 



and the same edge, this being square from the face. Apply the 
tips of the mould at the edge of the plank, as at a and o, (^,) and 
mark out the shape of the twist ; from a and o, draw the lines, a 
h and o c, across the edge of the plank, the angles, e a h and e o 
Cj corresponding with kfd at Fig. 277 ; turning the plank up as 
at Bj apply the tips of the mould at b and c, and mark it out as 
shown in the figure. In sawing out the twist, the saw must be 
be moved in the direction, a b ; which direction will be perpen- 
dicular when the twist is held up in its proper position. 

In sawing by the face-mould, the sides of the rail are obtained ; 
the top and bottom, or the upper and the lower surfaces, are ob- 
tained by squaring from the sides, after having bent the falling- 
mould around the outer, or convex side, and marked by its edges. 
Marking across by the ends of the falling-mould will give the 
position of the butt-joint. 

384. — Elucidation of the process by which the direction of 
the butt-joint is obtained in Art. 380. Mr. Nicholson, in his 
Carpenter^s Guide, has given the joint a different direction to 
that here shown ; he radiates it towards the centre of the cylin- 
der. This is erroneous — as can be shown by the following 
operation : 

In Fig. 280, a r j i'ls, the plan of a part of the rail about the 
joint, 5 w is the stretch-out of a i, and g p is the helinet, or ver- 
tical projection of the plan, a r j i, obtained according to Art 



328 



AMERICAN HOUSE-CARPENTER. 



gR 




Fig. 'iSO. 



382. Bisect r t, part of an ordinate from the centre of the plan, 
and through the middle, draw c b, at right angles to g v ; from 
b and c, draw c d and b e, at right angles to s u ; from d and e, 
draw lines radiating towards the centre of the plan : then d o 
and e m will be the direction of the joint on the plan, according to 
Nicholson, and c b its direction on the falling-mould. It will be 
admitted that all the lines on the upper or the lower side of the rail 
which radiate towards the centre of the cylinder, as c? o, e m or 
ij^ are level j for instance, the level line, w t?, on the top of the 



STAIRS. 



229 



rail in the helinet, is a true representation of the radiating line, j i 
on the plan. The line, b A, therefore, on the top of the rail in 
the helinet, is a true representation of e mon the plan, and A: c on 
the bottom of the rail truly represents d o. From k, draw k /, 
parallel to c b, and from h, draw hf, parallel to 6 c ; join I and 
b, also c and/; then c k I b will be a true representation of the 
end of the lower piece, B^ and c fh b of the end of the upper 
piece, A ; and/ k oi h I will show how much the joint is open on 
the inner, or concave side of the rail. 




Fig. '981. 



230 



AMERICAN HOUSE-CARPENTER. 



To show that the process followed in Art. 380 is correct, let d o 
and e m, {Fig. 281,) be the direction of the butt-joint found as at 
Fig. 275. Now, to project, on the top of the rail in the helinet, a 
line that does not radiate towards the centre of the cylinder, as j 
kj draw vertical lines from J and k to w and A, and join w and h ; 
then it will be evident that whissi true representation in the helinet 
of j k on the plan, it being in the same plane as ; k, and also in the 
same winding surface as w v. The Hue, I w, also, is a true repre- 
sentation on the bottom of the helinet of the line, j' A:, in the plan. 
The line of the joint, e m^ therefore, is projected in the same way 
and truly by i 6 on the top of the helinet ; and the line, d o, by 
c a on the bottom. Join a and i, and then it will be seen that 
the lines, c a, a i and i b, exactly coincide with c 6, the line of 
the joint on the convex side of the rail ; thus proving the lower 
end of the upper piece, A, and the upper end of the lower piece, 
B, to be in one and the same plane, and that the direction of the 
joint on the plan is the true one. By reference to Fig. 275, it will 
be seen that the line, I i, corresponds to a; i in Fig. 281 ; and 
that e k in that figure is a representation of/ b, and i k of d b. 





Fig.ses. 



In getting out the twists, the joints, before the falling-mould is 



STAIRS. 231 

applied, are cut perpendicularly, the face-mould being long enough 
to include the overplus necessary for a butt-joint. The face-mould 
for A, therefore, would have to extend to the line, i b ; and that for 
B, to the line, yz. Being sawed vertically at first, a section of the 
joint at the end of the face-mould for A, would be represented in 
the helinet by h if g. To obtain the position of the line, h i, on 
the end of the twist, draw i .«?, {Fig-. 282,) at right angles to if, 
and make i s equal to m e at Fig. 281 ; through 5, draw 5 g, pa- 
rallel to if, and make s h equal to 5 6 at Fig. 281 ; join h and i ; 
make i/equal to i /at Fig. 281, and from f, draw/^, parallel to i 
b ; then i b gf will be a perpendicular section of the rail over the 
line, e m, on the plan at Fig. 281, corresponding toi b gf in the 
helinet at that figure ; and when the rail is squared, the top, or 
back, must be trimmed off to the line, i b, and the bottom to the 
line, fg. 

385. — To grade the front string of a stairs, havi?ig winders 
in a quarter-circle at the top of the flight connected with flyers 
at the bottom. In Fig. 283, a b represents the line of the facia 
along the floor of the upper story, bee the face of the cylinder, 
and c d the face of the front string. Make g b equal to | of the 
diameter of the baluster, and draw the centre-line of the rail, y*^, 
g h i and ij, parallel to a b, b e c and c d; make g k and g I 
each equal to half the width of the rail, and through k and /, 
draw lines for the convex and the concave sides of the rail, parallel 
to the centre-line ; tangical to the convex side of the rail, and parallel 
to k m, draw no; obtain the stretch-out, g r, of the semi-circle, k 
J) m, according to Art. 92 ; extend ab to t, and k mio s ; make c s 
equal to the length of the steps, and i u equal to 18 inches, and de- 
scribe the arcs, s t and u 6, parallel to mp ; from t, draw t w, tend- 
ing to the centre of the cylinder ; from 6, and on the line, 6ux, run 
ofl" the regular tread, as at 5, 4, 3, 2, 1 and v ; make it x equal to 
half the arc, u 6, and make the point of division nearest to ar, as 
v, the limit of the parallel steps, or flyers ; make r o equal iomz ; 
from 0, draw o a^. at right angles to n o, and equal to one rise ; 



232 



AMERICAN HOUSE-CARPENTER. 



713 A3 



r8 h^ 




Fig. 283. 



from a^5 draw a^ s, parallel to it o, and equal to one tread ; from s, 
through 0, draw 5 6\ 

Then from w, draw w c^, at right angles to n o, and set up, on 
the line, w c', the same number of risers that the floor, J., is above 
the first winder, B, as at 1, 2, 3, 4, 5 and 6; through 5, (on the 
arc, 6 u,) draw dr e^, tending to the centre of the cylinder ; from 
c^, draw s^/^, at right angles to n o, and through 5, (on the line. 



STAIRS. 233 

w c',) draw ^'/^ parallel Xono ; through 6, (on the line, w c^,) 
and/^, draw the line, K^ ¥ ; make 6 coequal to half a rise, and 
from c' and 6, draw c^ r and 6/, parallel ion o ; make h^ i^ equal 
to h^p ; from i^, draw iH-*^, at right angles to i^ A^, and from/-, 
draw/*^^, at right angles to/Vi%' upon F, with Ic^ p for radius, 
describe the arc,/^ i%- make 6^ Z^ equal to h"^ f^-, and ease off the 
angle at h^ by the curve, f^ P. In the figure, the curve is de- 
scribed from a centre, but in a full-size plan, this would be imprac- 
ticable ; the best way to ease the angle, therefore, would be with 
a tanged curve, according to Art. 89. Then from 1, 2, 3 and 4, 
(on the line, w c^,) draw lines parallel to 7i o, meeting the curve in 
m^, 7i^, 0^ and p"^ ; from these points, draw lines at right angles to 
n 0, and meeting it in x"^, r^, s"^ and f; from x^ and r\ draw lines 
tending to w^, and meeting the convex side of the rail in y^ and 
z"^ ; make ni v^ equal to r s"^, and tu w'^ equal to r f ; from y-, z''. 
^;^ and iv"^, through 4, 3, 2 and 1, draw lines meeting the line of 
the wall-string in a^, 6^, c^ and cP ; from e^, where the centre-line of 
the rail crosses the line of the floor, draw e^f^, at right angles to n 
0, and from /^, through 6, draw/^ g"^ ; then the heavy lines,/^^', 
e^ (P, if' d^ z^ 6^, v^ &^ \c^ (P^ and zy^ will be the lines for the risers, 
which, being extended to the line of the front string, b e c d^ will 
give the dimensions of the winders, and the grading of the front 
string, as was required. 

386. — To obtain the falling-mould for the twists of the last- 
mentioned stairs. Make P g^ and P h^, {Fig. 283,) each equal 
to half the thickness of the rail ; through h^ and g^^ draw h^ r 
dind g^f, parallel to P s ; assuming k k^ and m w^ on the plan as 
the amount of straight to be got out with the twists, make 7i q 
equal to k k^, and r P equal to ?n m^ ; from n and Z^, draw lines at 
right angles to n o, meeting the top of the falling-mould in rv' and 
0^ ; from o^, draw a line crossing the falling-mould at right angles 
to a chord of the curve, /^ V ; through the centre of the cylinder, 
draw u^ 8, at right angles to 7i o : through 8, draw 7 9, tending to 
k^ ; then n^ 7 will be the falling-mould for the upper twist, and 7 

0^ the falling-mould for the lower twist. 

30 



234 



AMERICAN HOUSE-CARPENTER. 



387. — To obtain the face-moulds. The moulds for the twists 
of this stairs may be obtained as at Art. 380 ; but, as the faUing- 
mould in its course departs considerably from a straight line, it 
would, according to that method, require a very thick plank for 
the rail, and consequently cause a great waste of stuiF. In order, 
therefore, to economize the material, the following method is to 
be preferred — in which it will be seen that the heights are taken 
in three places instead of two only, as is done in the previous 
method. 




Fig, 284. o 



Case 1. — When the middle height is above a line joining 
the other two. Having found at Fig. 283 the direction of the 
joint, w s^ and p e, according to Art. 380, make k p e a^ {Fig. 
284,) equal to W p^ e p in Fig. 283 ; join h and c, and from o, 
draw A, at right angles to b c ; obtain the stretch-out of d g, as 
dfj and at Fig. 283, place it from the axis of the cylinder, p, to 
q^ ; from q^ in that figure, draw q^ r\ at right angles ion o ; also, 
at a convenient height on the line, n n% in that figure, and at 
right angles to that line, draw u^ v^ ; from b and c, in Fig. 284, 



STAIRS. 235 

draw h j and c Z, at right angles to b c ; make b j equal to u^ n' in 
Fig. 283, i h equal to lu^ r^ in th?it figure, and c Z equal \o v^ ^ \ 
from Zj through j*, draw / m ; from A, draw /i n, parallel to c b ; 
from 71. draw n r, at right angles to b c, and join ?' and 5 ; through 
the lowest corner of the plan, as />, draw v e, parallel to b c ; from 
a, e, II, p, k, t, and from as many other points as is tliought ne- 
cessary, draw ordinates to the base-line, v e, parallel to ?' s : 
through A, draw w x, at right angles to in I ; upon n, with r s for 
radius, describe an intersecting arc at x, and join n and x ; from 
the points at wliich the ordinates from the plan meet the base- 
line, V e, draw ordinates to meet the line, m /, at right angles to v 
e ; and from the points of intersection on in Z, draw correspond- 
ing ordinates, parallel to nx ; make the ordinates which are pa- 
rallel to n :r of a length corresponding to those Vv^hich are parallel 
to r s, and through the points thus found, trace the face-mould 
as required. 

Case 2. — When the middle height is below a line joining 
the other two. The lower twist in Fig. 283 is of this nature. 
The face-mould for this is found at Fig. 285 in a manner similar 
to that at Fig. 284. The heights are all taken from the top ot 
the falling-mould at Fig. 283 ; b j being equal to lo 6 in Fig. 283. 
i h equal to x^ if" in that figure, and cl\ol^ ol Draw a line 
through J and Z, and from A, draw h tz, parallel to 6 c ; from n. 
draw n i\ at right angles to b c, and join r and s ; then r s will be 
the bevil for the lower ordinates. From A, draw h x, at right an- 
gles toj I ; upon ?i, with 7' s for radius, describe an intersecting 
arc at x, and join n and x : then n x will be the bevil for the upper 
ordinates, upon which the face-mould is found as in Case 1. 

388. — Elucidation of the foregoing method. — This method 
of finding the face-moulds for the handrailing of winding stairs, 
being founded on principles which govern cylindric sections, may 
be illustrated by the following figures. Fig. 286 and 287 repre- 
sent solid blocks, or prisms, standing upright on a level base, b d : 
'he upper surface, j a forming oblique angles with the face, b I — 



236 



AMERICAN HOUSE-CARPENTER. 




Fig. 285. 



in Fig. 286 obtuse, and in Fig. 287 acute. Upon the base, de- 
scribe the semi-circle, h s c ; from the centre, i, draw i s, at right 
angles to 6 c ; from 5, draw 5 x, at right angles to e d, and from i, 
draw i h, at right angles to b c ; make i h equal to 5 x, and join 
h and x ; then, h and x being of the same height, the line, h x, 
joining them, is a level line. From A, draw h w, parallel to h c, 
and from?^, draw n r, at right angles tob c; join r and 5, also n 



STAIBS. 



237 





Fiff. 286. 



!■ ijr. vJr^7. 



and ^; then, n and x being of the same height, n x is a level hne ; 
and this line lying perpendicularly over r 5, n x and r s must be 
of the same length. So, all lines on the top, drawn parallel to n 
or, and perpendicularly over corresponding Unes drawn parallel to 
r 5 on the base, must be equal to those lines on the base ; and by 
drawing a number of these on the semi-circle at the base and 
others of the same length at the top, it is evident that a curve, j 
X Z, may be traced through the ends of those on the top, which 
shall lie perpendicularly over the semi-circle at the base. 

It is upon this principle that the process at Fig. 284 and 285 
is founded. The plan of the rail at the bottom of those figures 
is supposed to lie perpendicularly under the face-mould at the top ; 
and each ordinate at the top over a corresponding one at the base. 
The ordinates, n x and r 5, in those figures, correspond to n x 
and r 5 in Fig. 286 and 287. 

In Fig. 288, the top, e a, forms a right angle with the face, d 
c ; all that is necessary, therefore, in this figure, is to find a line 
corresponding to h x in the last two figures, and that will lie level 
and in the upper surface ; so that all ordinates at right angles to 
dron the base, will correspond to those that are at right angles 



238 



AMERICAN HOUSE-CARPENTER. 




F\-^ 2S8. r 



to e c on the top. This ekicidates Fig: 276 ; at which the lines, 
h 9 and i 8, correspond to h 9 and i 8 in this figure. 




Fig. 289. 



389. — To find the hevil for the edge of the plank. The 
plank, before the face-mould is applied, must be bevilled accord- 
ing to the angle which the top of the imaginary block, or prism, 
in the previous figures, makes with the face. This angle is de- 
termined in the following manner : draw w i, [Fig. 289,) at right 
angles to i s, and equal to w hat Fig. 284 ; make i s equal to t 5 in 
that figure, and join iv and s ; then sw p will be the bevil required 
m order to apply the face-mould at Fig. 284. In Fig. 285, the 
middle height being below the line joining the other two, the bevil 
is therefore acute. To determine this, draw i s, (Fig. 290,) at 



STAIRS. 



239 




Fig. 290. 



right angles to i p, and equal to i 5 in Fig". 285 ; make 5 w equal 
\o h w in Fig. 285, and join w and i ; then tv i p will be the 
bevil required in order to apply the face-mould at Fig. 285. Al- 
though the falling-mould in these cases is curved, yet, as the 
plank is sprung, or bevilled on its edge, the thickness necessary 
to get out the twist may be ascertained according to Art. 381 — 
taking the vertical distance across the falling-mould at the joints, 
and placing it down from the two outside heights in Fig. 284 or 
285. After bevilling the plank, the moulds are applied as at Art. 
383 — applying the pitch-board on the bevilled instead of a square 
edge, and placing the tips of the mould so that they will bear the 
same relation to the edge of the plank, as they do to the line, j /, 
in Fig. 284 or 285. 




Fig. 291. 



390. — To apply the moulds ivithout bevilling the plank. 
Make w p, {Fig. 291,) equal to w p at Fig, 289, and the angle, 
bed, equal to 6 j I in Fig. 284 ; make p a equal to the thick- 
ness of the plank, as w? a in Fig. 289, and from a draw a o, pa- 
rallel tow d ; from c, draw c e, at right angles to w d, and join e 



240 



AMERICAN HOUSE'CARPENTER. 



and h ; then the angle, b e o,ona square edge of the plank, hav- 
ing a Ime on the upper face at the distance, p a, in Fig. 289, at 
which to apply the tips of the mould — will answer the same pur- 
pose as bevilling the edge. 

If the bevilled edge of the plank, which reaches from p to w, 
is supposed to be in the plane of the paper, and the point, a, to 
be above the plane of the paper as much as a, in Fig. 289, is dis- 
tant from the line, w p ; and the plank to be revolved on p 6 as 
an axis until the line, p w, falls below the plane of the paper, and 
the line, p a, arrives in it ; then, it is evident that the point, c, 
will fall, in the line, c e, until it lies directly behind the point, e, 
and the line, b c, will lie directly behind b e. 

k 




Fig. 292. 



391. — To find the b evils for splayed work. The principle 
employed in the last figure is one that will serve to find the bevils 
for splayed work — such as hoppers, bread-trays, &c. — and a way 
of applying it to that purpose had better, perhaps, be introduced 
in this connection. In Fig. 292, a 6 c is the angle at which the 
work is splayed, and b c?, on the upper edge of the board, is at 
right angles to a b ; make the angle, /^^', equal to a b c, and 
from/, draw /A, parallel to e a; from 6, draw b o, at right an- 
gles Xoab ; through o, draw i e, parallel to c 6, and join e and 
d ; then the angle, a e d, will be the proper bevil for the ends from 
the inside, ox k d e from the outside. If a mitre-joint is re- 



STAIRS. 211 

quired, setfg, the thickness of the stuff on the level, from e to 
m, and join th and d ; then k d m will be the proper bevil for a 
mitre-joint. 

If the upper edges of the splayed work is to be bevilled, so as 
to be horizontal when the work is placed in its proper position, 
fgj, being the same as a 6 c, will be the proper bevil for that 
purpose. Suppose, therefore, that a piece indicated by the lines, 
k g, g-f and /A, were taken off; then a line drawn upon the 
bevilled surface from d, at right angles to k d, would show the 
true position of the joint, because it would be in the direction of 
the board for the other side ; but a line so drawn would pass 
through the point, o, — thus proving the principle correct. So, if 
a line were drawn upon the bevilled surface from d, at an angle 
of 45 degrees to k d, it would pass through the point, n. 

392. — Another method for face-'inoulds. It will be seen by 
reference to Art. 388, that the principal object had in view in the 
preparatory process of finding a face-mould, is to ascertain upon it 
the direction of a horizontal line. This can be found by a method 
diflferent from any previously proposed ; and as it requires fewer 
lines, and admits of less complication, it is probably to be preferred. 
It can be best introduced, perhaps, by the following explanation . 

In Fig, 293,^' d represents a prism standing upon a level base, 
h </, its upper surface forming an acute angle with the face. 
h /, as at Fig. 287. Extend the base line, h c, and the raking 
line,^' Z, to meet at/; also, extend e d and ^ a, to meet at k ; 
from /, through A*, draw / m. If we suppose the prism to stand 
upon a level floor, o f m^ and the plane, ; g a /, to be extended 
to meet that floor, then it will be obvious that the intersection 
between that plane and the plane of the floor would be in the line, 
f k ; and the line, /A:, being in the plane of the floor, and also in 
the inclined plane, J ^ A;/, any line made in the plane, ^'^ A*/, 
parallel to/ A:, must be a level line. By finding the position of a 
perpendicular plane, at right angles to the raking plane, jf k g, 
we shall greatly shorten the process for obtaining ordinates. 

31 



242 



AMERICAN HOUSE-CARPENTER. 




Fig. 293, 



This may be done thus : from/, draw/ o, at right angles iofm; 
extend e 6 to o, and^ /, to t ; from o, draw o t, at right angles to 
ofj and join t and/; then t of will be a perpendicular plane, at 
right angles to the inclined plane, t g kf; because the base of 
the former, o/ is at right angles to the base of the latter,/ ^*, both 
these lines being in the same plane. From 6, draw h p, at right 
angles to o/ or parallel iofm ; from/;, draw p q, at right angles 
to o/ and from q, draw a line on the upper plane, parallel to fin, 
or at right angles to tf; then this line will obviously be drawn 
to the point, ^*, and the line, g j, be equal top b. Proceed, in the 
same way, from the points, 5 and c, to find x and I. 

Now, to apply the principle here explained, let the curve, 65 c, 
{Fig. 294,) be the base of a cylindric segment, and let it be re- 
quired to find the shape of a section of this segment, cut by a 
plane passing through three given points in its curved surface : 
one perpendicularly over 6, at the h'eight, bj; one perpendicu- 
larly over 5, at the height, s x ; and the other over c, at the height, 
c I — these lines being drawn at right angles to the chord of the 
base, b c. Fromj, through Z, draw a line to meet the chord line 
extended to/; from 5, draw 5 A:, parallel to b /, and from x. 
draw X k, parallel to jf ; from/ through k, draw/m; then/w 
will be the intersecting line of ^he plane of the section with the 



STAIRS. 



213 




Fig. 294. 



plane of the base. This line can be proved to be the intersection 
of these planes in another way ; from 6, through 6\ and from^*, 
through .-r, draw lines meeting at rn ; then the point, tu, will be 
in the intersecting line, as is shown in the figure, and also at 
Fig. 293. 

From/, draw/p, at right angles to f m : from b and c, and 
from as many other points as is thought necessary, draw ordinates, 
parallel to fm; make p q equal to h j^ and join q and/; from 
the points at which the ordinates meet the line, qf. draw others 
at right angles to q f; make each ordinate at A equal to its cor- 
responding ordinate at C, and trace the curve, g n i, through the 
points thus found. 

Now it may be observed that A is the plane of the section, B 
the plane of the segment, corresponding to the plane, q p/ of 
Fig. 293, and C is the plane of the base. To give these planes 
their proper position, let A be turned on qf as an axis until it 



244 AMERICAN HOUSE-CARPENTER. 

Stands perpendicularly over the line, qf^ and at right angles to 
the plane, B ; then, while A and B are fixed at right angles, let 
B be turned on the line, ]> /, as an axis until it stands perpendicu- 
larly over p/, and at right angles to the plane, C ; then the plane, 
J, will lie over the plane, C, with the several lines on one corres- 
ponding to those on the other ; the point, i, resting at /, the point, 
^^, at x^ and g at j ; and the curve, g n i, lying perpendicularly 
over b s c — as was required. If we suppose the cylinder to be 
cut by a level plane passing through the point, Z, (as is done in 
finding a face-mould,) it will be obvious that lines corresponding 
to Q'/ and jo/ would meet in / ; and the plane of the section. A, 
the plane of the segment, B, and the plane of the base, C, would 
all meet in that point. 

393. — To find the face-mould for a limid-rail according to 
the principles explained in the previous article. In Fig. 295, 
a e cf is the plan of a hand-rail over a quarter of a cylinder ; and 
in Fig. 296, a b c d is the falling-mould ; / e being equal to the 
stretch-out of a df in Fig. 295. From c, draw c A, parallel to 
ef; bisect h c in i, and find a point, as 6, in the arc, df {Fig. 
295.) corresponding to i in the line, h c; from i, {Fig. 296,) to 
the top of the falling-mould, draw i j, at right angles to Ac; at Fig. 
295, from c, through 6, draw c g, and from b and c, draw bj and 
c k, at right angles to ^ c ; make c k equal io h g at Fig. 296, 
and bj equal to ij at that figure ; from k, through j, draw k g, 
and from^, through a, drsiw g p ; then ^p will be the intersecting 
line, corresponding to/m in Fig. 293 and 294 ; through e, draw 
p 6, at right angles to g j^, and from c, draw c q. parallel to g p ; 
make r q equal to h g at Fig. 296 ; join p and q^ and proceed as 
in the previous examples to find the face-mould, A. The joint 
of the face-mould, u v, will be more accurately determined by 
finding the projection of the centre of the plan, o, as at w ; 
joining 5 and w^ and drawing ii v, parallel to s w. 

It may be noticed that c k and b j are not of a length corres- 
ponding to the above directions : they are but^ the length given. 



iiiAlRfe. 



24S 




Fig. 295. 



246 



AMERICAN H0r5'Ji'''?ARPENTEIl. 




Fig. 296. 



The object of drawing these lines is to find the point, g-, and that 
can be done by taking any proportional parts of the lines given. 
as well as by taking the whole lines. For instance, supposing c 
k and h j to be the full length of the given lines, bisect one in i 
and the other in m; then a line drawn from m, through i, will 
give the point, g, as was required. The point, g, may also be 



STAIRS. 



247 



obtained thus : at Fig. 296, make h I equal to c 6 in Fig. 295 ; 
from Z, draw I k, at right angles to h c ; from 7, draw^* k, parallel 
to A c ; from g, through k, draw g n; at Fig. 295, make b g 
equal io In in Fig. 296 ; then g will be tlie point required. 

The reason why the points, a, b and c, in the plan of the rail a^ 
Fig. 295, are taken for resting points instead of e, i and/, is this : 
the top of the rail being level, it is evident that the points, a and e, 
in the section a e, are of the same height ; also that the point, i, is of 
the same height as 6, and c as/. Now, if a is taken for a point 
in the inclined plane rising from the line g ;:>, e must be below 
that plane ; if b is taken for a point in that plane, i must be below 
it ; and if c is in the plane,/ must be below it. The rule, then, 
for taking these points, is to take in each section the one that is 
nearest to the line, g p. Sometimes the line of intersection, g p, 
happens to come almost in the direction of the line, e r : in such 
case, after finding the line, see if the points from which the 
heights were taken agree with the above rule ; if the heights 
were taken at the wrong points, take them according to the rule 
above, and then find the true line of intersection, which will not 
vary nmch from the one already found. 




Fig. -297. 



394.— To appl]/ the face-mould thus found to the plank. 
The face-mould, when obtained by this method, is to be applied 
to a square-edged plank, as directed at Art. 383, with this differ- 
ence : instead of applying both tips of the mould to the edge of 



248 AMERICAN HOUSE-CARPENTER. 

the plank, one of them is to be set as far from the edge of the 
plank, as x^ in Fig. 295, is from the chord of the section p q — as 
is shown at Fig. 297. A. in this figure, is the mould applied on 
the upper side of the plank, i5, the edge of the plank, and C, the 
mould applied on the under side ; a b and c d being made equal 
to q X in Fig. 295, and the angle, e a c, on the edge, equal to the 
angle, p q r, at Fig. 295. In order to avoid a waste of stuff, it 
would be advisable to apply the lips of the mould, e and 6, im- 
mediately at the edge of the plank. To do this, suppose the 
moulds to be applied as shown in the figure ; then let A be re- 
volved upon e until the point, 6, arrives at g^ causing the line, e b, 
to coincide with eg: the mould upon the under side of the 
plank must now be revolved upon a point that is perpendicularly 
beneath e, as /; from/, draw / A, parallel to i d. and from (/, 
draw d A, at right angles to i d ; then revolve the mould, C, upon 
/, until the point. A, arrives at J, causing the line,/ A, to coincide 
with/^', and the line, i d, to coincide with k I j then the tips of 
the mould will be at k and I. 

The rule for doing this, then, will be as follows : make the an- 
gle, ^/A', equal to the angle q v x, at Fig. 295 ; make /A' equal 
to/i, and through A-, draw k /, parallel to ij ;. then apply the 
corner of the mould, i, at /r, and the other corner c?, at the line, k I. 
The thickness of stuff is found as at Art. 381. 
395. — To regulate the application of the falling-mould. 
Obtain, on the line, h c, [Fig. 296,) the several points, r, q,2h I 
and m, corresponding to the points, J^, a^^ z^ y, &c., at Fig. 295 ; 
from r q Pj (fee, draw the lines, r t, q ii, p v, (fee, at right angles 
to he; make A .«?, r t, q u, &c., respectively equal to 6 c"; r q, 5 
d', &c., at Fig. 295 ; through the points thus found, trace the 
curve, s w c. Then get out the piece, g s c, attached to the fall- 
ing-mould at several places along its length, as at z, z, z, &c. 
In applying the falling-mould with this strip thus attached, the 
edge, sw c, will coincide with the upper surface of the rail piece 



STAIRS. 



249 



before it is squared ; and thus show the proper position of the fall 
ing-mould along its whole length. (See Art. 403.) 

SCROLLS FOR HAND-RAILS. 

396. — General rule for finding the size a7id jjosition of the 
regulating square. The breadth which the scroll is to occupy, 
the number of its revolutions, and the relative size of the reguJa 
ting square to the eye of the scroll, being given, multiply tlie 
number of revolutions by 4, and to the product add the number 
of times a side of the square is contained in the diameter of th^ 
eye, and the sum will be the number of equal parts into which 
the breadth is to be divided. Make a side of the regulating 
square equal to one of these parts. To the breadth of the scroll 
add one of the parts thus found, and half the sum will be the 
length of the longest ordinate. 



















6 








5 
























8 










I- 






















4 




1 















Fiff. 298. 



397. — To find the proper centres in the regulating square. 
Let a 2 1 6, {Fig. 29S,) be the size of a regulating square, found 
according to the previous rule, the required number of revolu- 
tions being If. Divide two adjacent sides, as a 2 and 2 1, into 
as many equal parts as there are quarters in the number of revo- 
lutions, as seven ; from those points of division, draw lines across 
the square, at right angles to the lines divided ; then, 1 being the 
first centre, 2, 3, 4, 5, 6 and 7, are the centres for the other quar- 
ters, and 8 is the centre for the eye ; the heavy lines that deter- 

32 



250 



AMERICAN HOUSE-CARPENTER. 



mine th«se centres being each one part less in length than its pre- 
ceding line. 

^ M 




Fig. 299. 



398. — To describe the scroll for a hand-rail over a curtail 
step. Let a b, (Fig. 299,) be the given breadth. If the given 
number of revolutions, and let the relative size of the regulating 
square to the eye be ^ of the diameter of the eye. Then, by the 
rule, If multiplied by 4 gives 7, and 3, the number of times a 
side of the square is contained in the eye, being added, the sum 
is 10. Divide a 6, therefore, into 10 equal parts, and set one from 
b to c ; bisect a c in e ; then a e will be the length of the longest 
ordinate, (1 c/ or 1 e.) From a, draw a d, from e, draw e 1, and 
from b, draw 6/, all at right angles to a b ; make e 1 equal to e 
a, and through 1, draw 1 d, parallel to a b ; set b c from 1 to 2, 
and upon 1 2, complete the regulating square ; divide this square 
as at Fig. 298 ; then describe the arcs that compose the scroll, as 
follows: upon 1, describe d e; upon 2, describe e f; upon 3., 
describe/^ ; upon 4, describe g A, &c. ; make d I equal to the 



STAIRS. 



251 



width of the rail, and upon 1, describe Im ; upon 2, aescribe m 
w, <fec. ; describe the eye upon 8, and the scroll is completed. 

399. — To describe the scroll for a curtail step. Bisect d I, 
{Fig". 299,) in o, and make o v equal to ^ of the diameter of a 
baluster ; make v w equal to the projection of the nosing, and e 
X equal to w I; upon 1, describe w y, and upon 2, describe y z , 
also upon 2, describe x i ; upon 3, describe i j, and so around to 
z ; and the scroll for the step will be completed. 

400. — 7b determine the 'position of the balusters under the 
scroll. Bisect c?Z, [Fig. 299,) in o, and upon 1, with 1 o for ra- 
dius, describe the circle, o r u ; set the baluster at p fair with the 
face of the second riser, c'^ and from p, with half the tread in the 
dividers, space off as at o, q^ r, 5, ^, u^ &c., as far as q^ ; upon 2, 
3, 4 and 5, describe the centre-line of the rail around to the eye 
of the scroll ; from the points of division in the circle, o r ?/, draw 
lines to the centre-line of the rail, tending to the centre of the 
eye, 8 ; then, the intersection of these radiating lines with the 
centre-line of the rail, will determine the position of the balusters, 
as shown in the figure. 




Fijr. 300. 



401. — To obtain the falling-mould for the raking part of the 
scroll. Tangical to the rail at h, {Fig. 299,) draw h k, parallel to d 
a; then k d^ will be the joint between the twist and the other part 
of the scroll. Make d e^ equal to the stretch-out of d e, and upon d 



252 



AMERICAN HOUSE-CARPENTER. 



e'^, find the position of the point, k^ as at ^ ; at Fig. 300, make e d 
equal to e^ d in Fig. 299, and d c equal to d & in that figure ; 
from c, draw c a, at right angles to e c, and equal to one rise ; 
make c h equal to one tread, and from 6, through a, draw 6 j , 
bisect a c in Z, and through Z, draw ?7i 9-, parallel to e h ; m q is 
the height of the level part of a scroll, which should always be 
about 3A feet from the floor ; ease off the angle, m/J, according 
to Art. 89, and draw g w n, parallel to m x j^ and at a distance 
equal to the thickness of the rail ; at a convenient place for the 
joint, as i, draw i n, at right angles to b j ; through n, draw / A, 
at right angles to c h ; make d k equal to d k"^ in Fig. 299, and 
from A*, draw k 0, at right angles to e h ; at Fig. 299, make d 
h^ equal to c? ^ in Fig. 300, and draw h^ 6^, at right angles to d 
h^ ; then k a^ and h^ i^ will be the position of the joints on the 
plan, and at Fig. 300, p and i n, their position on the falling- 
mould ; and p i ?i, {Fig. 300,) will be the falling-mould re- 
quired. 




Fig. SOL 



402. — To describe the face-mould. At Fig. 299, from/:, draw 
k r'^, at right angles to r'- d ; at Fig. 300, make h r equal to h"^ r^ 
in Fig. 299, and from r, draw r s, at right angles to r h ; from 
the intersection of r 5 with the level line, m q^ through ^, draw s 
t ; at Fig. 299, make K^ b'^ equal to ^ ^ in Fig. 300, and join b"^ 
and r'^ ; from a^, and from as many other points in the arcs, a^ I 
and k d, as is thought necessary, draw ordinates to r'^ d, at right 
angles to the latter ; make r b, {Fig. 301,) equal in its length and 
in its divisions to the line, r"^ b% in Fig. 299 ; from r, n, 0, /?, q 



STAIRS. 



253 



and Z, draw the lines, r k, n d, o a, p e, ^/and I c, at right an- 
gles to r b, and equal to r^ k, d' s\ f a\ &c., in Fig. 299 ; 
through the points thus found, trace the curves, A; Z and a c, and 
complete the face-mould, as shown in the figure. This mould is 
to be applied to a square-edged plank, with the edge, I 6, parallel 
to the edge of the plank. The rake lines upon the edge of the 
plank are to be made to correspond to the angle, s t h, in Fig. 
300. The thickness of stuff required for this mould is shown at 
Fig. 300, between the lines s t and u v—u v being drawn pa- 
rallel to s t. 

403.— All the previous examples given for finding face-moulds 
over winders, are intended for ?noulded rails. For roimd rails, 
the same process is to be followed with this difference : instead 
of working from the sides of the rail, work from a centre-line. 
After finding the projection of that line upon the upper plane, 
describe circles upon it, as at Fig. 262, and trace the sides of the 
moulds by the points so found. The thickness of stuff for the 
twists of a round rail, is the same as for the straight; and the 
twists are to be sawed square through. 




g 

Fig. 302. 



254 



AMERICAN HOUSE-CARPENTER 



404. — To ascertain the form of the newel-cap from a section 
of the rail. Draw a b, {Fig. 302,) through the widest part of 
the given section, and parallel io c d ; bisect a 6 in e, and through 
a, e and b, draw h i,fg, and kj, at right angles to a b ; at a con- 
venient place on the line, fg^ as o, with a radius equal to half 
the width of the cap, describe the circle, i j g ; make r I equal 
to e b or e a ; join I and j, also I and i ; from the curve, / b, to 
the line, I j, draw as many ordinates as is thought necessary, 
parallel to f g ; from the points at which these ordinates meet 
the line, I j, and upon the centre, o, describe arcs in continuation to 
meet op; from n, t, x, &c., draw n s, t u, (fee, parallel to f g ; 
make n s, t u, &c., equal to e f lo v^ &c. ; make x y, &c., equal 
to z dy &c. ; make o 2, o 3, &:c., equal to on, o t, &,c. ; make 2 4 
equal to n s, and in this way find the length of the lines crossing 
m ; through the points thus found, describe the section of the 
newel-cap, as shown in the figure. 




Fig. 303. 



405. — To find the true position of a butt joint for the twists of 
a moulded rail over platform stairs. Obtain the shape of the 
mould according to Art. 373, and make the line a b, Fig. 303, 
equal to a c, Fig. 269 ; from b, draw b c, at right angles to a b, 
and equal in length to n ??i, Fig. 269 ; join a and c, and bisect a c 
in ; through o draw e f at right angles to a c, and d k, parallel 
to cb ; make o d and o k each equal to half e h at Fig. 269 ; 
through e and /, draw h i and g j, parallel to a c. At Fig. 270, 
make n a equal to e d. Fig. 303, and through a, draw r p, at right 
angles to n c ; then r p will be the true position on the face-mould 
for a butt joint, as was required. The sides must be sawn verti- 



STAIRS. 



255 



cally as described at Art. 374, but the joint is to be sawn square 
through the plank. The moulds obtained for round rails, (Art. 
371,) give the line for the joint, when applied to either side of the 
plank ; but here, for moulded rails, th'; line for the joint can be 
obtained from only one side. Whe i the rail is canted up, the 
joint is taken from the mould laid on the upper side of the lower 
twist, and on the under side of the upper twist ; but when it is 
canted down, a course just the reverse of this is to be pursued. 
When the rail is not canted, either up or dowr?. the vertical joint, 
obtained as at A7't. 373, will be a butt joint, and therefore, in such 
a case, the process described in this article will be unnecessary 



NOTE TO ARTICLE 369. 



Platform stairs vrith a large cylinder. Instead of 
placing ihe platform-risers at the spring of the cyl- 
inder, a more easy and graceful appearance may be 
given to the rail, and the necessity of canting either 
of the twists entirely obviated, by fixing the place of 
the above risers at a certain distance within the cyl- 
inder, as shown in the annexed cut — the lines indi- 
cating the face of the risers cutting the cylinder at k 
and I, instead of at p and q, the spring of the cylin- 
der. To ascertain the position of the risers, let a 6 c 
be the pitch-board of the lower flight, and c d e that 
of the upper flight, these being placed so that b c 
and c d shall form a right line. Extend a c to cut 
de in f; draw / g parallel to d b, and of indefinite 
length : draw g o at right angles to / g, and equal 
in length to the radius of the circle formed by the 
centre of the rail in passing around the cylinder; 
on o as centre describe the semicircle j g i ; make 
h equal to the radius of the cylinder, and describe 
on o the face of the cylinder p h q ; then extend d b 
across the cylinder, cutting it in I and k — giving the 
position of the face of the risers, as required. To 
find the face-mould for the twists is simple and ob- 
vious : it being merely a quarter of an ellipse, hav- 
ing j for semi-minor axis, and the distance on the 

rake corresponding to o g, on the plan, for the semi-major axis, found thus,— extend i j to 
meet a /, then from this point of meeting to / is the semi-major axis. 




SECTION VIL— SHADOWS. 



406. — The art of drawing consists in representing solids upon 
a plane surface : so that a curious and nice adjustment of lines is 
made to present the same appearance to the eye, as does the 
human figure, a tree, or a house. It is by the effects of light, in 
its reflection, shade, and shadow, that the presence of an object is 
made known to us ; so, upon paper, it is necessary, in order that 
the delineation may appear real, to represent fully all the shades 
and shadows that would be seen upon the object itself. In this 
section I propose to illustrate, by a few plain examples, the simple 
elementary principles upon which shading, in architectural sub- 
jects, is based. The necessary knowledge of drawing, prelim- 
inary to this subject, is treated of in the Introduction, from Art. 
1 to 14. 

407. — TTie inclination of the line of shadow. This is always, 
in architectural drawing, 45 degrees, both on the elevation and the 
plan ; and the sun is supposed to be behind the spectator, and 
over his left shoulder. This can be illustrated by reference to 
Fig. 304, in which A represents a horizontal plane, and B and C 
two vertical planes placed at right angles to each other. A rep- 
resents the plan, C the elevation, and B a vertical projection 
from the elevation. In finding the shadow of the plane, By the 



SHADOWS. 



257 




Fig. 304. 



line, a b, is drawn at an angle of 45 degrees with the horizon, and 
the hne, c b, at the same angle with the vertical plane, B. The 
plane, B, being a rectangle, this makes the true direction of the 
sun's rays to be in a course parallel to d b ; which direction has 
been proved to be at an angle of 35 degrees and 16 minutes with 
the horizon. It is convenient, in shading, to have a set-square 
with the two sides that contain the right angle of equal length ; 
this will make the two acute angles each 45 degrees ; and will 
£^ive the requisite bevil when worked upon the edge of the T- 
square. One reason why this angle is chosen in preference to 
another, is, that when shadows are properly made upon the draw- 
ing by it, the depth of every recess is more readily known, since 
the breadth of shadow and the depth of the recess will be equal. 

To distinguish between the terms shade and shadow, it will be 
understood that all such parts of a body as are not exposed to the 
direct action of the sun's rays, are in shade ; while those parts 
which are deprived of light by the interposition of other bodies, 
are in shadow. . 

33 



258 



AMERICAN HOUSE-CARPENTER. 




Fig. 305. 



Fig. 306. 





Fig. 307. 



Fig. 308. 



408. — To find the line of shadow on mouldings and other ho- 
rizontaUy straight projections. Fig. 305, 306, 307 and 308, 
lepresent various mouldings in elevation, returned at the left, in 
the usual manner of mitreing around a projection. A mere in- 
spection of the figures is sufficient to see how the line of shadow 
is obtained ; bearing in mind that the ray, a h, is drawn from the 
projections at an angle of 45 degrees. Wliere there is no return 
at the end, it is necessary to draw a section, at any place in the 
length of the mouldings, and find the line of shadow from that. 

409. — To find the line of shadoio cast by a shelf. In Fig. 309, 
A is the plan, and B is the elevation of a shelf attached to a wall. 
From a and c, draw a b and c d, according to the angle previously 
directed : from 6, erect a perpendicular intersecting c ddX d ; from 
d, draw d e, parallel to the shelf ; then the lines, c d and d e, will 
define the shadow cast by the shelf. There is another method of 
finding the shadow, without the plan, A. Extend the lower line 
of me shelf to/, and make c/ equal to the projection of the shelf 



SHADOWS. 



25^ 




Fig. 309. 

from the wall ; from/, draw/^, at the customary angle, and from 
c, drop the vertical line, c g, intersecting f g ^X g ; from g^ draw 
g e, parallel to the shelf, and from c, draw c d, at the usual angle ; 
then the lines, c d and d e, will determine the extent of the shadow 
as before. 









1 


e 






iir '" 1 




1 


''i'iiiiiiiililllli!iii||i{;ii 


C 


, 


lliijiiiiiiiini 


iiiiiiiiiiiiiiiiiii 


liilllitllllllllll 






/ 


y^b 








c 

Fig. 310 




- 



B 



410. — To find the shadow cast hy a shelf , which is wider at 
one end than at the other. In Fig. 310, A is the plan, and B 
the elevation. Find the point, d, as in the previous example, and 
from any other point in the front of the shelf, as a, erect the perpen- 
dicular, a e ; from a and e, draw a h and e c, at the proper angle, 
and from 6, erect the perpendicular, h c, intersecting e c m c ; 



^260 



AMERICAN nOUSE-CARPENTEll. 



from d, through c, draw d o ; then the hnes, i d and d o, will give 
the limit of the shadow cast by the shelf. 




Fig. 311. 

411. — To find the shadow of a shelf having one end acute or 
obtuse angled. Fig. 311 shows the plan and elevation of an 
acute-angled shelf. Find the line, e g, as before ; from a, erect 
the perpendicular, ab ; join 6 and e ; then b e and e g will define 
the boundary of shadow. 




Fig. 312. 

412. — To find the shadow cast by an inclined shelf. In Fig. 
312, the plan and elevation of such a shelf is shown, having also 
one end wider than the other. Proceed as directed for finding 
the shadows of Fig, 310, and find the points, d and c ; then a d 
and d c will be the shadow required. If the shelf had been 



SHADOWS. 



261 



parallel in width on the plan, then the line, d c, would have been 
parallel with the shelf, a b. 














/ 










1 
d 








\l 1 1 illi ililll! 


D 






e 


c 



Fig. 313. 



Fig. 314. 



413. — To find the shadow cast hy a shelf inclined in its ver- 
tical section either upward or downward. From a, {Fig, 313 
and 314,) draw a b, at the usual angle, and from b, draw b c, 
parallel with the shelf; obtain the point, e, by drawing a line 
from c?, at the usual angle. In Fig. 313, join e and i ; then i e 
and e c will define the shadow. In Fig. 314, from o, draw o i, 
parallel with the shelf; join i and e ; then i e and e c will be the 
shadow required. 

The projections in these several examples are bounded by- 
straight lines ; but the shadows of curved lines may be found in 
the same manner, by projecting shadows from several points in 
the curved line, and tracing the curve of shadow through these 
points. Thus — 





Fig. 316. 



Fig. 315. 



262 



AMERICAN HOTjSE-CARPENTEU. 



414. — To find the shadow of a shelf having its front edge^ or 
endy curved on the plan. In Fig. 315 and 316, A and A show an 
example of each kind. From several points, as a, a, in the plan, 
and from the corresponding points, o, o, in the elevation, draw 
rays and perpendiculars intersecting at e, e, &c. ; through these 
points of intersection trace the curve, and it will define the shadow. 



e 


^.v 


e 


11 

e 


lliiiiiiiiiiiiiiiii 


11^ 


lllllllllllllll;! 


iiiiiiiiiiiiiiii 




/ 







Fig. 31- 



415. — To find the shadow of a shelf curved in the elevation. 
In Fig. 317, find the points of intersection, e, e and e, as in the 
last examples, and a curve traced through them will define the 
shadow. 

The preceding examples show how to find shadows when cast 
upon a vertical plarie ; shadows thrown upon curved surfaces are 
ascertained in a similar manner. Thus — 




Fig. 318. 



SHADOWS. 



263 



416. — To find the shadow cast upon a cylindrical wall by a 
projection of any kind. By an inspection of Fig. 318, it will be 
seen that the only difference between this and the last examples, 
is, that the rays in the plan die against the circle, a b, instead of 
a straight line. 




Fig. 319. 

417. — To find the shadow cast by a shelf upon an inclined 
loaU. Cast the ray, a b, {Fig. 319,) from the end of the shelf to 
the face of the wall, and from b, draw b c, parallel to the shelf; 
cast the ray, d e, from the end of the shelf ; then the lines, d e 
and e c, will define the shadow. 

These examples might be multiplied, but enough has been 
given to illustrate the general principle, by which shadows in all 
instances are found. Let us attend now to the application of this 
principle to such familiar objects as are likely to occur in practice. 




264 



AMERICAN IiaUSE-CARPENTER. 



418. — To find the shadow of a projecting horizontal becmi. 
From the points, a, a, 6cc., (Fig. 320,) cast rays upon the wall , 
the intersections, e, e, e, of those rays with the perpendiculars 
drawn from the plan, will define the shadow. If the beam be in- 
clined, eitlier on the plan or elevation, at any angle other than a 
light angle, the difference in the manner of proceeding can be seen 
by reference to the preceding examples of inclined shelves, &c- 




Fig. 321. 

419. — To find the shadow in a recess. From the point, a, 
[Fig. 321,) in the plan, and h in the elevation, draw the ra3'S, a c 
and h e ; from c, erect the perpendicular, c e, and from e, draw 
the horizontal line, e d; then the lines, c e and e d, will show tlie 
extent of the shadow. This applies only where the back of the 
recess is parallel with the face of the wall. 




FiL'. 32^. 



420. — To find the shadow in a recess, when the face of the 
wall is inclined, and the hack of the recess is vertical. In Fig 
322, A shows the section and B the elevation of a recess of this 



SHADOWS. 



265 



kind. From h, and from any other point in the Hne, h a, as a, 
draw the rays, h c and a e ; from c, a, and e, draw the horizontal 
lines, eg", a f, and eh; from J and /, cast the rays, d i and/ A ; 
from i, through h^ draw i s ; then s i and i g will define the 
shadow. 

d 




mmi, 



Fig. 323. 



421. — To find the shadow in a fireplace. From a and h, 
{Fig. 323,) cast the rays, a c and b e, and from c, erect the per- 
pendicular, c e ; from e, draw the horizontal line, e o, and join o 
and d ; then c e, e o, and o d, will give the extent of the shadow. 



^miiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii|{ 
"11 s 



iiiiiinimiiii !iiMiiiiiiiii!iii;i! 



Fig. 324. 

422. — To find the shadow of a moulded window-lintel. Cast 
rays from the projections, a, o, &c., in the plan, {Fig. 324,) and 
d, e, &c., in the elevation, and draw the usual perpendiculars in- 
tersecting the rays at ^, i, and i ; these intersections connected 

34 



206 



AMERICAN HOUSE-CARPENTER. 



and horizontal lines drawn from them, will define the shadow. 
The shadow on the face of the lintel is found by casting a ray 
back from i to s, and drawing the horizontal line, s n. 




Fig. 325. 

423. — To find the shadow cast hy the nosing of a step. From 
«, {Fig. 325,) and its corresponding point, c, cast the rays, a h 
and c d, and from h, erect the perpendicular, h d ; tangical to the 
curve at e, cast the ray, e /, and from e, drop the perpendicular, 
e o, meeting the mitre-line, a g, in o ; cast a ray from o to z, and 
from i, erect the perpendicular, i f ; from h, draw the ray, h k; 
from f to d and from d to k, trace the curve as shown in the 
figure ; from k and h, draw the horizontal lines, k n and h s ; then 
the limit of the shadow will be completed. 

424. — To find the shadow thrown by a pedestal upon steps. 
From a, {Fig. 326,) in the plan, and from c in the elevation, draw 
the rays, a b and c e ; then a o will show the extent of the shadow 
on the first riser, as at ii ; f g will determine the shadow on the 
second riser, as at ^ ; c d gives the amount of shadow on the 
first tread, as at C, and h i that on the second tread, as at D ; 
which completes the shadow of the left-hand pedestal, both on the 
plan and elevation. A mere inspection of the figure will be suf- 



SHADOWS. 



267 




[ll!l!l||ll 



iiiiiiiiiiiiijiiiiiiiiiiiiiiiiiiimiiiiiii!iii!iiiiiiiiiiiiiii 



D 



T 



Miiiiiiiiinmi::'''' '!;''^!':'!TJiiiii|iiii 



Fig. 326. 

ficient to show how the shadow of the right-hand pedestal is 
obtained. 





Fig. 327. Fig. 3-i8, 

425. — To find the shadow thrown on a column by a square 
abacus. From a and b, {Fig. 327,) draw the rays, a c and b e, 
and from c, erect the perpendicular, c e ; tangical to the curve at 
c/, draw the ray, d f, and from It, corresponding to / in the plan, 
draw the ray, h o ; take any point between a and/, as i, and from 
this, as also from a corresponding point, ?2, draw the rays, i r and 
n s ; from r, and from d, erect the perpendiculars, r s and d o : 
through the points, e, s, and o, trace the curve as shown in the 
figure : then the extent of the shadow v/ill be defined. 

426. — To find the shadow thrown on a column by a circular 
abacus. This is so near like the last example, that no explanation 
will be necessary farther than a reference to the preceding article 



268 



AMERICAN HOUSE-CARPENTEK. 




Fisr. 329. 



427. — To find the shadows on the capitcd of a column. This 
may be done according to the principles explained in the examples 
already given ; a quicker way of doing it, however, is as follows. 
If we take into consideration one ray of light in connection with 
all those perpendicularly under and over it, it is evident that these 
several rays would form a vertical plane, standing at an angle of 
45 degrees with the face of the elevation. Now, we may sup- 
pose the column to be sliced, so to speak, with planes of this 



5?I1AD0\YS. 



269 



IH.UI 



I 1 1 I 1 1 1 1 I ' I 

II II I I I M I I I I I I 1 
M I M I I M I 1 I I I I I I I I ; 1 I ' 




FiK. 330. 



nature — cutting it in the lines, a h, c d, &c., {Fig. 329,) and, m 
tlie elevation, find, by squaring up from the plan, the lines of sec- 
tion which these planes would make thereupon. For instance : 
in finding upon the elevation the line of section, a Z), the plane 
cuts the ovolo at e, and therefore /will be the corresponding point 
upon the elevation ; h corresponds with g, i with^, o with s, and 
I with h. Now, to find the. shadows upon this line of section, cast 
from m, the ray, r/i n, from h, the ray, h o, &c. ; then that part of 
the section indicated by the letters, m f i n, and that part also be- 
tween h and o, will be under shadow. By an inspection of the 
figure, it will be seen that the same process is applied to eacli line 
of section, and in that way the points, p, r, t, u, v, tv, x, as also 
1, 2, 3, &c., are successively found, and the lines of shadow 
traced through them. 

Fig. 330 is an example of the same capital with all the shadows 
finished in accordance with the lines obtained on Fig. 329. 

428. — To find the shadow thrown on a vertical wall by a 
column and entablature standing in advance of said wall. Cast 



270 



AMERICAN HOUSE-CARPENTER. 




Fig. 331. 



rays from a and h, {Fig. 331,) and find the point, c, as in tlie 
previous examples ; from d^ draw the ray, d e, and from e, the 
horizontal line, e f ; tangical to the curve at g and A, draw the 
rays, g j and h z, and from i and j^ erect the perpendiculars, i I 
and^' k ; from m and n^ draw the rays, m/and n k^ and trace the 
curve between k and /; cast a ray from o to p, a. vertical line 
from p to 5, and through s, draw the horizontal line, s t ; the 
shadow as required will then be completed. 



SHADOWS. 



271 




Fig. 332. 



Fig. 332 is an example of the same kind as the last, with all 
the shadows filled in, according to the lines obtained in the pre- 
ceding figure. 





Fig. 333. 



429. — Fig. 333 and 334 are examples of the Tuscan cornice. 
The manner of obtaining the shadows is evident. 



272 



A.MERICAN HOUSE-CARPENTER. 




Fig. 334. 

430. — Reflected light. In shading, the finish and Ufe of an 
object depend much on reflected light This is seen to advantage 
in Fig. 330 and on the column in Fig. 332. Reflected rays are 
thrown in a direction exactly the reverse of direct rays ; therefore, 
on that part of an object which is subject to reflected light, the 
shadows are reversed. The fillet of the ovolo in Fig. 330 is an 
example of this. On the right-hand side of the column, the face 
of the fillet is much darker than the cove directly under it. The 
reason of this is, the face of the fillet is deprived both of direct 
and reflected light, whereas the cove is subject to the latter. 
Other instances of the effect of reflected light will be seen in the 
other examples. 



APPENDIX 



GLOSSARY. 



Tenns not found here can be found in the lists of definitions m other parts of this booV.i 

or m fommon dictionaries. 



Abacus. — The uppermost member of a capital. 

Ahhatoir. — A slaughter-house. 

Abbey. — The residence of an abbot or abbess. 

Abutment. — That part of a pier from which the arch springs. 

Acanthus. — A plant called in English, beards-breech. Its leaves are 
employed for decorating the Corinthian and the Composite capitals. 

Acropolis. — The highest part of a city ; generally the citadel. 

Acroteria. — The small pedestals placed on the extremities and apex 
of a pediment, originally intended as a base for sculpture. 

Aisle. — Passage to and frorp the pews of a church. In Gothic ar- 
chitecture, the lean-to wings on the sides of the nave. 

Alcove. — Part of a chamber separated by an esirade, or partition of 
columns. Recess with seats, &c., in gerdens. 

Altar. — A pedestal whereon sacrifice was offered. In modern 
churches, the area within the railing in front of the pulpit. 

Alto-relievo. — High i-elief ; sculpture projecting from a surface so as 
to appear nearly isolated. 

Amphitheatre —A double theatre, employed by the ancients for the 
exhibition of gladiatorial fights and other shows. 

Ancones. — Trusses employed as an apparent support to a cornice 
upon the flanks o'' the architrave. 

A.nnulet. — A small square moulding used to separate others ; the 
fill^'ts in the Doric capital under the ovolo, and those which separate 
tiiG flutings of columns, are known by this term. 

Antce. — A pilaster attached to a wall. 

Apiary. — A place for keeping beehives. 

Arabesque. — A building after the Arabian style. 

Areostyle. — An intercolumniation of from four to five diameters. 

Arcade — A series of arches. 

Arch. — An arrangement of stones or other material in a curvilinear 
form, so as to perform the office of a lintel and carry superincumbent 
weights. 

Architrave. — That part of the entablature which rests upon the 
capital of a column, and is beneath the frieze. The casing and 
mouldings about a door or window. 



4 APPENDIX. 

Archivolt. — The ceiling of a vault : the under surface of an arch. 

Area. — Superficial measurement. An open space, below the level 
of the ground, in front of basement windows. 

Arsenal. — A public establishment for the deposition of arms and 
warlike stores. 

Astragal. — A small moulding consisting of a half-round with a fillet 
on each side. 

Attic. — A low story erected over an order of architecture. A low 
additional story immediately under the roof of a building. 

Aviary. — A place for keeping and breeding birds. 

Balcony. — An open gallery projecting from the front of a building. 

Baluster. — A small pillar or pilaster supporting a rail. 

Balustrade. — A series of balusters connected by a rail. 

Barge-course. — That part of the covering which projects over the 
gable of a building. 

Base. — The lowest part of a wall, column, &c. 

Basement-story. — That which is immediately under the principal 
story, and included within the foundation of the building. 

Basso-relievo. — Low relief; sculptured figures projecting from a 
surface one-half their thickness or less. See Alto-relievo. 

Battering. — See Talvs. 

Battlement. — Indentations on the top of a wall or parapet. 

Bay-window. — A window projecting in two or more planes, and not 
forming; the segment of a circle. 

Bazaar. — A species of mart or exchange for the sale of various ar- 
ticles of merchandise. 

Bead. — A circular moulding. 

Bed-mouldings. — Those mouldings which are between the corona 
and the frieze. 

Belfry. — That part of a steeple in which the bells are hung : an- 
ciently called camjMnile. 

Belvedere. — An ornamental turret or observatory commanding a 
pleasant prospect. 

Boiv-ivindow. — A window projecting in curved lines. 

Bressummer. — Abeam or iron tie supporting a wall over a gateway 
or other opening. 

Brick -Hogging. — The brickwork between studs of partitions. 

Buttress. — A projection from a wall to give additional strength. 

CaMe. — A cylindrical moulding placed in flutes at the lower part of 
the column. 

Camber. — To give a convexity to the upper surface of a beam. 

Campanile. — A tower for the reception of bells, usually, in Italy, 
separated from the church. 

Canopy. — An ornamental covering over a seat of state. 

Cantalivers. — The ends of rafters under a projecting roof. Pieces 
of wood or stone supporting the eaves. 

Capital. — The uppermost part of a column included between the 
shaft and the architrave. 



APPENDIX. 5 

Carmmnsera. — In the East, a large public building for the reception 
of travellers by caravans in the desert. 

Carpentry. — (From the Latin, carpentum, carved wood.) That de- 
partment of science and art which treats of the disposition, the con- 
struction and the relative strength of timber. Th^ first is called de- 
scriptive, the second constructive, and the last mechanical carpentry. 

Caryatides. — Figures of women used instead of columns to support 
an entablature. 

Casino. — A small country-house. 

Castellated. — Built with battlements and turrets in imitation of an- 
cient castles. 

Castle. — A building fortified for military defence. A house with 
towers, usually encompassed with walls and moats, and having a don- 
jon, or keep, in the centre. 

Catacombs. — Subterraneous places for burying the dead. 

Cathedral. — The principal church of a province or diocese, wherein 
the throne of the archbishop or bishop is placed. 

Cavetto. — A concave moulding comprising the quadrant of a circle. 

Cemetery. — An edifice or area where the dead arc interred. 

Cenotaph. — A monument erected to the memory of a person buried 
in another place. 

Centring. — The temporary woodwork, or framing, whereon any 
vaulted work is constructed. 

Cesspool. — A well under a drain or pavement to receive the waste- 
water and sediment. 

Chamfer. — The bevilled edge of any thing originally right-angled. 

Chancel. — That part of a Gothic church in which the altar is placed. 

Chantry. — A little chapel in ancient churches, with an endowment 
for one or more priests to say mass for the relief of souls out of purga- 
tory. 

Chapel. — A building for religious worship, erected separately from 
a church, and served by a chaplain. 

Chaplet. — A moulding carved into beads, olives, &c. 

Cincture. — The ring, listel, or fillet, at the top and bottom of a co- 
lumn, which divides the shaft of the column from its capital and base. 

Circus. — A straight, long, narrow building used by the Romans for 
the exhibition of public spectacles and chariot races. At the present 
day, a building enclosing an arena for the exhibition of feats of horse- 
manship. 

Clerestory. — The upper part of the nave of a church above the 
roofs of the aisles. 

Cloister. — The square space attached to a regular monastery or 
large church, having a peristyle or ambulatory around it, covered with 
a range of buildings. 

Coffer-dam. — A case of piling, water-tight, fixed in the bed of a 
river, for the purpose of excluding the water while any work, such as 
a wharf, wall, or the pier of a bridge, is carried up. 

Collar-beam. — A horizontal beam framed between two principal 
rafters above the tie-beam. 

Collonade. — A range of columns. 

Columbarium. — A pigeon-house. 



G APPENDIX. 

Column. — A vertical, cylindrical support under the entablature of 
an order. 

Common-rafters. — The same as jack-rafters, which see 

Conduit. — A long, narrow, walled passage underground, for secret 
communication between different apartments. A canal or pipe for the 
conveyance of water. 

Conservatory. — A building for preserving curious and rare exotic 
plants. 

Consoles. — The same as ancones, which see. 

Contour. — The external lines which bound and terminate a figure. 

Convent. — A building for the reception of a society of religious per- 
sons. 

Coping. — Stones laid on the top of a wall to defend it from the 
weather. 

Corbels. — Stones or timbers fixed in a wall to sustain the timbers of 
a floor or roof. 

Cornice. — Any moulded projection which crowns or finishes the 
part to which it is affixed. 

Corona. — That part of a cornice which is between the crown- 
moulding and the bed-mouldings. 

Cornucopia. — The horn of plenty. 

Corridor. — An open gallery or communication to the difforent aparr- 
ments of a house. 

Cove. — A concave moulding. 

Cripple-rafters. — The short rafters which are spiked to the hip-rafter 
of a roof. 

Crockets. — In Gothic architecture, the ornaments placed along the 
angles of pediments, pinnacles, &c. 

Crosettes. — The same as ancones, which see. 

Crypt. — The under or hidden part of a building. 

Culvert. — An arched channel of masonry or brickwork, built be- 
neath the bed of a canal for the purpose of conducting water under it. 
\ny arched channel for water underground. 

Cupola. — A small building on the top of a dome. 

Curtail -step. — A step with a spiral end, usually the first of the flight. 

Cusps. — The pendents of a pointed arch. 

Cyma. — An ogee. There are two kinds ; the cyma-recta, having 
the upper part concave and the lower convex, and the cyma-revrsa. 
with the upper part convex and the lower concave. 

Dado. — The die, or part between the base and cornice of a pedestal. 

Dairy. — An apartment or building for the preservation of milk, and 
the manufacture of it into butter, cheese, &c. 

Dead-shoar. — A piece of timber or stone stood vertically in brick- 
work, to support a superincumbent weight until the brickwork which 
is to carry it has set or become hard. 

Decastyle. — A building having ten columns in front. 

Dentils. — (From the Latin, denies, teeth.) Small rectangular blocks 
used in the bed-mouldings of some of the orders. 

Diaslyle. — An intercolumniation of three, or, as some say, four 
diameters. 



APPENDIX. ' 

Die. — That part of a pedestal included between the base and the 
cornice ; it is also called a dado. 

Dodecastyle. — A building having twelve columns in front. 

Donjon. — A massive tower within ancient castles to which the gar- 
rison might retreat in case o^' necessity. 

Dooks. — A Scotch term given to wooden bricks. 

Dormer. — A window placed on the roof of a house, the frame being 
placed vertically on the rafters. 

Dormitory. — A sleeping-room. 

Dovecote. — A building for keeping tame pigeons. A columbarium. 

Echinus. — The Grecian ovolo. 

Elevation. — A geometrical projection drawn on a plane at right an- 
gles to the horizon. 

Entallature. — That part of an order which is supported by the co- 
lumns ; consisting of the architrave, frieze, and cornice. 

Eustyle. — An intercolumniation of two and a quarter diameters. 

Exchange. — A building in which merchants and brokers meet to 
transact business. 

Extrados. — The exterior curve of an arch. 

Facade. — The principal front of any building. 

Face-mould — The pattern for marking the plank, out of which hand- 
railing is to be cut for stairs, &c. 

Facia, or Fascia. — A flat member like a band or broad fillet. 

Falling-mould. — The mould applied to the convex, vertical surface 
of the rail-piece, in order to form the back and under surface of the 
rail, and finish the squaring. 

Festoon. — An ornament representing a wreath of flowers and leaves. 

Fillet. — A narrow flat band, listel, or annulet, used for the separa- 
tion of one moulding from another, and to give breadth and firmness 
:.o the edges of mouldings. 

Flutes. — Upright channels on the shafts of columns. 

Flyers. — Steps in a flight of stairs that are parallel to each other. 

Forum. — -In ancient architecture, a public market ; also, a place 
where the common courts were held, and law pleadings carried on. 

Foundry. — A building in which various metals are cast into moulde 
or shapes. 

Frieze. — That part of an entablature included between the archi- 
trave and the cornice. 

Gahle. — The vertical, triangular piece of wall at the end of a rooi. 
from the level of the eaves to the summit. 

Gain. — A recess made to receive a t^non or tusk. 

Gallery. — A common passage to several rooms in an upper storv. 
A long room for the reception of pictures. A platform raised on co- 
lumns, pilasters, or piers. 

Girder. — The principal beam in a floor for supporting the binding 
and other joists, whereby the bearing or length is lessened. 

Glyph. — A vertical, sunken channel. From their number, those in 
the Doric order are called triglyphs. 



B APPENDIX. 

Granary. — A building for storing grain, especially that intended to 
be kept for a considerable time. 

Groin. — The line formed by the intersection of two arches, which 
cross each other at any angle. 

Gv.it(E. — The small cylindrical pendent ornaments, otherwise called 
drops, used in the Doric order under the triglyphs, and also pendent 
from the mutuli of the cornice. 

Gymnasium. — Originally, a space measured out and covered with 
sand for the exercise of athletic games : afterwards, spacious buildings 
devoted to the mental as well as corporeal instruction of youth. 

Hall. — The first large apartment on entering a house. The public 
room of a corporate body. A manor-house. 

Ham. — A house or dwelling-place. A street or village : hence 
Notting/ia77i, Bucking^a7?i, &c. Hamlet, the diminutive of ham, is a 
small street or village. 

Helix. — The small volute, or twist, under the abacus in the Corin- 
thian capital. 

Hem. — The projecting spiral fillet of the Ionic capital. 

Hexastyle. — A building having six columns in front. 

Hip-rafter. — A piece of timber placed at the angle made by two ad- 
jacent inclined roofs. 

Homestall. — A mansion-house, or seat in the country. 

Hotel, or Hostel. — x\ large inn or place of public entertainment. A 
large house or palace. 

Hot-house. — A glass building used in gardening. 

Hovel. — An open shed. 

Hut. — A small cottage or hovel generally constructed of earthy 
materials, as strong loamy clay, &c. 

Impost. — The capital of a pier or pilaster which supports an arch. 
Intaglio. — Sculpture in which the subject is hollowed out, so that 
the impression from it presents the appearance of a bas-relief. 
Intercolumniation. — The distance between two columns. 
Intrados. — The interior and lower curve of an arch. 

Jack-rafters. — Rafters that fill in between the principal rafters of a 
roof; called also common-rafters. 

Jail. — A place of legal confinement. 

Jamhs. — The vertical sides of an aperture. 

Joggle-piece. — A post to receive struts. 

Joists. — The timbers to which the boards of a floor or the laths of a 
ceiling are nailed. 

Keep. — The same as donjon, which see. 
Key-stone. — The highest central stone of an arch. 
Kiln. — A building for the accumulation and retention of heat, in or- 
der to dry or burn certain materials deposited within it. 
King-post. — The centre-post in a trussed roof. 
Knee, — A convex bend in the back of a hand-rail. See Ramp. 



APPENDIX. V 

Lactarium. — The same as dairy, which see. 

Lantern. — A cupola having windows in the sides for lighting an 
apartment beneath. 

Larmier. — The same as corona, which see. 

Lattice. — A reticulated window for the admission of air, rather than 
light, as in dairies and cellars. 

Lever -hoards. — Blind-slats : a set of boards so fastened that they 
may be turned at any angle to admit more or less light, or to lap upon 
each other so as to exclude all air or light through apertures. 

Lintel. — A piece of timber or stone placed horizontally over a door, 
window, or other opening. 

Listel. — The same as fillet, which see. 

Lobby. — An enclosed space, or passage, communicating with the 
principal room or rooms of a house. 

Lodge. — A small house near and subordinate to the mansion. A 
cottage placed at the gate of the road leading to a mansion. 

Loop. — A small narrow window. Loophole is a term applied to the 
vertical series of doors in a warehouse, through which goods are de- 
livered by means of a crane. 

Luffer -boar ding. — The same as lever-boards, which see. 

Luthern. — The same as dormer, which see. 

Mausoleum, — A sepulchral building — so called from a very cele- 
brated one erected to the memory of Mausolus, king of Caria, by his 
wife Artemisia. 

Metopa. — The square space in the frieze between the triglyphs of 
the Doric order. 

Mezzanine. — A story of small height introduced between two of 
greater height. 

Minaret. — A slender, lofty turret having projecting balconies, com- 
mon in Mohammedan countries. 

Minster. — A church to which an ecclesiastical fraternity has been 
or is attached. 

Moat. — An excavated reservoir of water, surrounding a house, cas- 
tle or town. 

ModilUon. — A projection under the corona of the richer orders, re- 
sembling a bracket. 

Module. — The semi-diameter of a column, used by the architect as 
a measure by which to proportion the parts of an order. 

Monastery. — A building or buildings appropriated to the reception of 
monks. 

Monopteron. — A circular collonade supporting a dome without an 
enclosing wall. 

Mosaic. — A mode of representing objects by the inlaying of small 
cubes of glass, stone, marble, shells, &c. 

Mosque. — A Mohammedan temple, or place of worship. 

Mullions. — The upright posts or bars, which divide the lights in u 
Gothic window. 

Muniment-house. — A strong, fire-proof apartment for the keeping 
and preservation of evidences, charters, seals, &c., called muniments. 

1* 



10 APPENDIX. 

Museum. — A repository of natural, scientific and literary, curiosities, 
or of works of art. 

Mutule. — A projecting ornament of the Doric cornice supposed to 
represent the ends of rafters. 

Nave. — The main body of a Gothic church. 

Newel. — A post at the starting o:" landing of a flight of stairs. 

Niche. — A cavity or hollow place in a wall for the reception of a 
statue, vase, &;c. 

Nogs. — Wooden bricks. 

Nosing. — The rounded and projecting edge of a step in stairs. 

Nunnery. — A building or buildings appropriated for the reception of 
nuns. 

Obelisk. — A lofty pillar of a rectangular form. 

Octastyle. — A building with eight columns in front. 

Odemn. — Among the Greeks, a species of theatre wherein the poets 
and musicians rehearsed their compositions previous to the public pro- 
duction of them. 

Ogee. — See Cyma. 

Orangery. — A gallery or building in a garden or parterre fronting 
the south. 

Oriel-window. — A large bay or recessed window in a hall, chapel, or 
other apartment. 

Ovolo. — A convex projecting moulding whose profile is the quad- 
rant of a circle. 

Pagoda. — A temple or place of worship in India. 

Palisade. — K fence of pales or stakfes driven into the ground. 

Parapet. — A small wall of any material for protection on the sides 
of bridges, quays, or high buildings. 

Pavilion. — A turret or small building generally insulated and com- 
prised under a single roof. 

Pedestal. — A square foundation used to elevate and sustain a co- 
lumn, statue, &c. 

Pediment. — The triangular crowning part of a portico or aperture 
which terminates vertically the sloping parts of the roof : this, in 
Gothic architecture, is called a gable. 

Penitentiary. — A prison for the confinement of criminals whose 
crimes are not of a very heinous nature. 

Piazza. — A square, open space surrounded by buildings. This 
term is often improperly used to denote a 'portico. 

Pier. — A rectangular pillar without any regular base or capital. 
The upright, narrow portions of walls between doors and windows are 
known by this term. 

Pilaster. — A square pillar, sometimes insulated, but more common 
ly engaged in a wall, and projecting only a part of its thickness. 

PUes. — Large timbers driven into the ground to make a secure 
foundation in marshy places, or in the bed of a river. 

Pillar. — A column of irregular form, always disengaged, and al- 



APPENDIX. -1 

ways deviating from the proportions of the orders ; whence the distinc- 
tion between a pillar and a column. 

Pinnacle. — A small spire used to ornament Gothic buildings. 

Planceer. — The same as soffit, which see. 

Phnth. — The lower square member of the base of a column, pedes- 
tal, or wall. 

Porch. — An exterior appendage to a building, forming a covered 
appioach to one of its principal doorways. 

Portal.— The arch over a door or gate ; the framework of the gate ; 
the lesser gate, when there are two of different dimensions at one en- 
trance. 

Portcullis. — A strong timber gate to old castles, made to slide up 
and down vertically. 

Portico. — A colonnade supporting a shelter over a walk, or ambu- 
latory. 

Priory. — A building similar in its constitution to a monastery or 
abbey, the head whereof was called a prior or prioress. 

Prism. — A solid bounded on the sides by parallelograms, and on the 
ends by polygonal figures in parallel planes. 

Prostyle. — A building with columns in front only. 

Purlines. — Those pieces of timber which lie under and at right an- 
gles to the rafters to prevent them from sinking. 

Pycnostyle. — An intercolumniation of one and a half diameters. 

Pyramid. — A solid body standing on a square, triangular or poly- 
gonal basis, and terminating in a point at the top. 

Quarry. — A place whence stones and slates are procured. 

Quay. — (Pronounced, key.) A bank formed towards the sea or on 
the side of a river for free passage, or for the purpose of unloading 
merchandise. 

Quoin. — An external angle. See Rustic quoins. 

Rabbet, or Rebate. — A groove or channel in the edge of a board. 

Ramp. — A concave bend in the back of a hand-rail. 

Rampant arch. — One having abutments of different heights. 

Regula. — The band below the tsenia in the Doric order. 

Riser. -^\n stairs, the vertical board forming the front of a step. 

Rostrum. — An elevated platform from which a speaker addresses an 
audience. 

Rotunda. — A circular building. 

Rubble-wall. — A wall built of unhewn stone. 

Rudenture. — The same as cable, which see. 

Rustic quoins. — The stones placed on the external angle of a build- 
ing, projecting beyond the face of the wall, and having their edges 
bevilled. 

Rustic-work. — A mode of building masonry wherein the faces of the 
stones are left rough, the sides only being wrought smooth where the 
union of the stones takes place. 



12 APPENDIX. 

Salon, or Saloon. — A lofty and spacious apartment comprehending 
the height of two stories with two tiers of windows. 

Sarcophagus. — A tomb or coffin made of one stone. 

Scantling. — The measu/e to which a piece of timber is to be or has 
been cut. 

. Scarfing. — The joining of two pieces of timber by bolting or nailing 
transversely together, so that the two appear but one. 

Scotia. — The hollow moulding in the base of a column, between the 
fillets of the tori. 

Scroll. — A carved curvilinear ornament, somewhat resembling in 
profile the turnings of a ram's horn. 

Sepulchre. — A grave, tomb, or place of interment. 

Sewer. — A drain or conduit for carrying off soil or water from any 
place. 

Shaft. — The cylindrical part between the base and the capital of a 
column. 

Shoar. — A piece of timber placed in an oblique direction to support 
a building or wall. 

Sill. — The horizontal piece of timber at the bottom of framing ; the 
timber or stone at the bottom of doors and windows. 

Soffit — The underside of an architrave, corona, &c. The underside 
of the heads of doors, windows, &c. 

Summer. — The lintel of a door or window ; a beam tenoned into a 
girder to support the ends of joists on both sides of it. 

Systyle. — An intercolumniation of two diameters. 

TcBnia. — The fillet which separates the Doric frieze from the archi- 
trave. 

Talus. — The slope or inclination of a wall, among workmen called 
battering. 

Terrace. — An area raised before a building, above the level of the 
ground, to serve as a walk. 

Tesselated pavement. — A curious pavement of Mosaic work, com- 
posed of small square stones. 

Tetrastyle. — A building having four columns in front. 

Thatch. — A covering of straw or reeds used on the roofs of cottages, 
barns, &c. 

Theatre. — A building appropriated to the representation of drama..c 
spectacles. 

Tile. — A thin piece or plate of baked clay or other material used for 
the external covering of a roof. 

Tomh. — A grave, or place for the interment of a human body, in- 
cluding also any commemorative monument raised over such a place. 

Torus. — A moulding of semi-circular profile used in the bases of 
columns. 

Tower. — A lofty building of several stories, round or polygonal. 

Transept. — The transverse portion of a cruciform church. 

Transom. — The beam across a double-lighted window ; if the win. 
dow have no transom, it is called a clerestory window. 



APPENDIX. 13 

Tread. — That part of a step which is included between the face of 
its riser and that of the riser above. 

Trellis. — A reticulated framing made of thin bars of wood for 
screens, windows, &c. 

Triglyph. — The vertical tablets in the Doric frieze, chamfered on 
.he two vertical edges, and having two channels in the middle. 

Tripod. — A table or seat with three legs. 

Trochilus. — The same as scotia, which see. 

Truss. — An arrangement of timbers for increasing the resistance to 
cross-strains, consisting of a tie, two struts and a suspending-piece. 

Turret. — A small tower, often crowning the angle of a wall, &c. 

Tusk — A short projection under a tenon to increase its strength. 

Tympanum. — The naked face of a pediment, included between the 
level and the raking mouldings. 

Underpinning. — The wall under the ground-sills of a building. 
University. — An assemblage of colleges under the supervision of a 
senate, &;c. 

Vault. — A concave arched ceiling resting upon two opposite paral- 
lel walls. 

Venetian-door. — A door having side-lights. 

Venetian-window. — A window having three separate apertures. 

Veranda. — An awning. An open portico under the extended roof 
of a building. 

Vestibule. — An apartment which serves as the medium of commu- 
nication to another room or series of rooms. 

Vestry. — An apartment in a church, or attached to it, for the pre- 
servation of the sacred vestments and utensils. 

Villa. — A country-house for the residence of an opulent person. 

Vinery. — A house for the cultivation of vines. 

Volute. — A spiral scroll, which forms the principal feature of the 
Ionic and the Composite capitals. 

Voussoirs. — Arch-stones 

Wainscoting. — Wooden lining of walls, generally in panels. 

Water-tahle. — The stone covering to the projecting foundation or 
other walls of a building. 

Well. — The space occupied by a flight of stairs. The space left 
beyond the ends of the steps is called the well-hole. 

Wicket. — A small door made in a gate. 

Winders. — In stairs, steps not parallel to each other. 

Zophorus. — The same as frieze, which see. 

Zyitos. — Among the ancients, a portico of unusual length, common 
Iv appropriated to gymnastic exercises. 



TABLE OF SaUARES, CUBES, AND ROOTS. 

(From Hutton's Mathematics.) 



No. 


1 Square. 


! Cube. 


' Sq. Root. CubeRoot. 


No. 
68 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


1 


1 


1 


1-0000000 1000000' 


4624 


314432 


8-2462113 


4-081655 


2 


4 


8 


l-4142136j 1-259921! 


69 


4761 


323509 


1 8-3066239 4 101566 


3 


9 


27 


1-73205081 1-442250 


70 


4900 


343000 


8-3666003 4-121285 


4 


16 


64 


1 20000000; 1-587401 


71 


5041 


357911 


8-4261498 4-140818 


5 


25 


125 


1 2-23606801 1-709976 


72 


5184 


373-248 


8-4852814 4-160168 


6 


35 


216 


2 -44948971 1-817121! 


73 


5329 


389017 


8-5440037 4-179339 


7 


49 


343 


: 2-6457513^ 1-912931 i 


74 


5476 


405224 


8-6023253 4-198336 


8 


64 


512 


; 2-82842711 2-000000: 


75 


5625 


421875 


8-6602540 4 217163 


9 


81 


729 


i 30000000J 2-080084' 


76 


5776 


438976 


1 8-7177979 4^35824 


10 


100 


1000 


{ 3-162-2777i 2-154435 


77 


5929 


456533 


8-7749644 4-254321 


11 


121 


1331 


3-3 166243; 2-223980! 


78 


6084 


474552 


8-8317609 4-272659 


12 


144 


1728 


3-4641016 2-289429 


79 


6241 


493039 


8-8381944 4-29t3840 


13 


169 


2197 


3-6055513 2351335 


80 


6400 


512000 


8-944-2719! 4-30H869 


14 


196 


2744 


: 3-7416574 2-410142 


81 


6561 


531441 


9-0000000 


i-'s^S'lVJ 


15 


225 


3375 


; 3-8:29833 2-466212 


82 


6724 


551368 


9-0553851 


4-344481 


16 


256 


4096 


4-00;)0000 


2-519342 


83 


6839 


571787 


9-1104336 


4-36-2071 


17 


289 


4913 


41231056 


2-571232 


84 


7056 


592704 


9-1651514 


4-379519 


18 


324 


5832 


4-2426407 


2-620741 


85 


7225 


6141-25 


9-2195445 


4-396830 


19 


361 


6859 


4-358-i989 


2-668402 


86 


7396 


636056 


9-2736185 


4-414005 


20 


400 


8000 


4-4721360 


2-714418 


87 


7569 


658503 


9-3273791 


4-131048 


21 


441 


9261 


4-58-25757 


2-758924 


88 


7744 


631472 


9-3808315 


4-447960 


22 


484 


10648 


4-6901158 2-802039 


89 


7921 


704969 


9-4339811 


4-464745 


23 


529 


12167 


4-7958315 


2-843S67 


90 


8100 


729000 


9-4888330 


4-481405 


24 


576 


13824 


4-8939795 


2-881499!, 


91 


8-281 


753571 


9-5393920 


4-497941 


25 


625 


15625 


5-0000000 


2-924018 


92 


8464 


778638 


9-5916630 


4 514357 


26 


676 


17576 


5-0990195 


2-962496 


93 


8649 


8-04357 


9-6436508 


4-6^0655 


27 


729 


19683 


5 196 1524 


3 000000 


94 


8836 


830534 


9-6953597 


4 546836 


28 


784 


21952 


5 29151)26 


3036589 


95 


9025 


857375 


9-7467943 


4-562903 


29 


841 


24389 


5-3351648 


3072317:' 


96 


9216 


884736 


9-7979590 


4-578357 


30 


900 


27000 


5-4772256 


3-107-232! 


97 


9409 


912673 


9-8483578 


4-59i701 


3i 


961 


29791 


5 5677644 


31413ji; 


98 


9604 


941192 


9-8994949 


4-G.10436 


32 


1024 


3-2768 


5-6568542 


3-174802 : 


99 


9301 


970299 


9-9493744 


4-625065 


33 


1089 


35937 


5-7445626 


3-207534 


100 


10000 


1000000 


10-0000000! 4 641589 


^ 


1156 


39304 


5-8309519 


3 239612 


101 


10201 


1030301 


10-04937561 4-657009 


35 


1225 


42875 


5-9160798 


3-271066 


102 


10404 


1061-208 


10-09950491 4-672329 


36 


1296 


- ,. - ^ 

46656 


6 0000000 


3-301927, 


103 


10609 


1092727 


10- 14889 1G[ 4-687548 


37 


1369 


50653 


0-OS27625 


3-332222; 


104 


10816 


1124864 


10-1980390' 4-702669 


38 


1444 


54S72 


6-1644140 


3-361975 


105 


110-25 


1157625 


10-2469508^ 4-717694 


'^•j 

*; 


1521 


59319 


6-24 19980 3-391211 


106 


11236 


1191016 


10-2956301: i -732623 


40 


1600 


61000 


6-3245553 3 419952 


107 


11449 


1225043 


10-314080'i; 4-747459 
10-3923048 4-76-2203 


i^ 


1681 


68921 


6-4U31212 3 U8217 


108 


11664 


1259712 


42 


1764 


74088 


6-4807407 3-476027i, 


109 


11381 


1295029 


10-4403065' -,-776856 


43 


1319 


79507 


6-5574385 3-503398 


110 


12100 


1331000 


10-4880835 4-7914-20 


44 


1936 


85184 


6-6332496 3-53;J3l8 


111 


12321 


1367631 


10-53565381 4-805895 


45 


2025 


91125 


6-7032039 3-556893 


112 


12544 


1404928 


10-583O052' 4 •820-284 


46 


2116 


97335 


6-7823300 3-583J48^ 


113 


12769 


144-2897 


10-63014581 4-834538 


4; 


2209 


103823 


6-8556548 3-(k)8S2"j 


114 


12996 


1431514 


10-67707.^^3 4-8iS308 


48 


2304 


1 10592 


6-9282032 3-634241 


115 


13-225 


1520875 


10 7-238053: 4-862944 


49 


2401 


117649 


7-0000000 3-659306 


116 


13455 


1560896 


10-77032v^6; 4-876999 


5U 


2500 


125000 


7-0710678 3-6 S 1031 


117 


13689 


1601613 


10-816!;53.Si 4-890973 


51 


26U1 


132651 


7-1414284 3-70843 j' 


118 


13924 


1643032 


10-8627805' 4-904868 


52 


2704 


140608 


7-2111026 3-732511 


119 


14161 


1685159 


10-9087121J 4-918635 
10-9544512 4-9324-24 


53 


2309 


14S877 


7-2S01099 3-756285 


120 


14400 


1728000 


54 


2916 


157464 


7-3481692 3779763 i 


121 


14641 


1771561 


11-0000001) 4-946037 


55 


3025 


166375 


7-4161985 3-502952 


122 


14884 


1815348 


ll-0453i>lO 4'959676 


56 


3136 


175616 


7-483314S 3-825862i 


123 


15129 


1860867 


11-0905365 4-973190 


57 


3249 1 


185193 


7-5498344 3-848501! 


124 


15376 


19066-24 


11-1355-287 4-986631 


58 


3364 


195112 


7-6157731 


3-8708771 


125 


156-25 


1953125 


11-1803399 5-000000 


59 


3481 


205379 


7-68114571 


3-89-2996 1 


1-26 


15376 


2000376 


11-2249723 5-013298 


60 


3600 


216000 


7-7459667, 


3-914868! 


127 


16129 


2048383 


11-2694277 


5-026526 


61 


3721 


226981 


7-8102497i 


3-936497ii 


128 


16334 


2097152 


11-3137085 


5 039681 


62 


3^44 


238328 


7-87400791 


3-957891!' 


129 


16641 


2146689 


11-3578167 


5-052774 


63 


3969 


250047 


7-9372539' 


3-979057!' 


130 


16900 


2197000 


11-4017543 


5 065797 


64 


40% 


262144 


8-0000000 


4-000000! i 


131 


17161 


224Sim\ 


1 J -4455231 


5-07y753 


65 


4^25 


274r)25 


8-06225771 


4-020726! 


132 


17424 


2299968; 


n-4891253 


5-0916i3 


66 


4356 


2874'j6 


8-1240384 


4-041-240 1 


133 


17689! 


2352637 


11.53-25626 


5-104469 


67 


4489 


300763 


8-1853528 


4-061548 1 


134 


17956' 


2406104 


11-57583691 


511T280 



APPENDIX. 



15 



No. 


Square. 


Cube. 


Sq. Root. CubeRooU 


No. 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


135 


18225 


2160375 


11-6189500 5-129928 


202 


4U804 


8242408 


14-2126704 


5-867464 


136 


18496 


2515456 


11-661903S 5-142563 


203 


41209 


8365427 


14-2478068 


5-877131 


137 


18769 


2571353 


11-7046999 


5-155137 


204 


41616! 84896641 14-2o23569l 5-836765| 


133 


39044 


2628072 


11-7473401 


5-167649 


205 


42025 


8615125 


1 14-31782111 5-3963681 


139 


19321 


2635619 


11-7893-261 


5-180101 


206 


4243t 


8741816 


1 14-3527001 


5-905941 


140 


19600 


2744000 


11-8321596 


5-19-2494 


207 


42849 


8869743 


1 14-3874946 


5-915482 


141 


li)881 


2803221 


11-8743422 


5 •2043-28 


208 


43264 


8998912! 14-4222051! 5 9249921 


142 


20164 


2S63288 


11-9163753 


5-217103 


209 


43681 


9129329 


1 14-4568323! 5-934473 


143 


20449 


2924207 


11-9532607 


5-229321 


210 


44100 


9-261000 


14-4913767 5943922 


144 


2,)736 


2985934 


12-0000000 


5-241483 


211 


44521 


9393931 


1 14-52583901 5-953342 


145 


al025 


3043625 


12 0415946 


5-253533 


212 


44944 


9528128 


I 14-5602198 5-96-2732 


146 


21316 


3112136 


12-0830460 


5-265637 


213 


45369 


9663597 


1 14-5945195 5-972093 


147 


21609 


31715523 


12-1243557 


5-277632! 214 


45796 


9300344 


i 14-6-287338! 5-931424 


148 


21904 


3211792 


12-1655251 


5-239572 215 


46225 


9933375 


14-6623783: 5990726 


149 


22201 


3307949 


12-2065556 


5 301459 1 216 


46656 


10077696 


14-6969385; 6-000000 


150 


22500 


3375000 


12-2174437 


5-313293i 217 


47089 


10218313 


14-7309199 


6-009245 


151 


22S01 


3442951 


12-2832057 


5-325074 218 


47524 


10^360232 


14-7613231 


6-013462 


152 


23104 


3511803 


12-3238280 


5-336803 219 


47961 


10503459 


14-7986486 


6-027650 


153 


23409 


3581577 


12-3693169 


5-318481' 220 


48400 


10643000 


14-8323970 


6-036811 


154 


23716 


3652264 


12-4096736 


5-36010S:j 221 


43S41 


10793861 


14-8660687i 6-045943 


155, 


24025 


3723375 


12-449399.'- 


5-37I6S5i 222 


49284 


10941048 


14-8996644! 6-055049 


156 


2433?i 


3796416 


12-4399J60 


5-333213 


; 223 


49729 


11089567 


14-93318451 6064127 


157 


24649 


3869393 


12-5299641 


5-394691; 


224 


50176 


1 1239424 


14-9666295 6073178 


158; 


24964 


3944312 


12-5698051 


5-406120i 


225 


50625 


11390625 


15-0000000 6-082202 


159! 


25281 


4019679 


12-6095202 


5-417501 


226 


51076 


11513176 


150332964! 6-091199 


160; 


25600 


4096000 


12-6491106 


5 ■428335; 1 227 


51529 


11697083 


15-0665192 6-100170 


16i 


25921 


4173281 


12-6335775 


5 -440 1-22: 223 


51984 


11852352 


15-09966391 6-109115 


1621 


25244 


4251523 


12-7279-221 


5-451362: 229 


5-2441 


12008939 


151327460 6-118033 


163' 


26569 


4330747 


12-7671153 


5-46-2556 230 


52900 


12167000 


I51657509i 6-126926 


164j 


26896 


4410944 


12-8062485 


5-473704 i 231 


53361 


12326391 


15- 193634-2! 6-135792 


165 


27225 


4492125 


12-8452326 


5-434807:; 232 


53324 


12487163 


15-2315462! 6-144634 


166 


S7556 


4574296 


12-8840987 


5-495365 ; 233 


51-289 


12649337 


15-2643375 6-153449 


1671 


27SS9 


4657463 


12 9228480 


5-506878 i 234 


54756 


12812904 


15-2J70585' 6-16-2240 


1631 


28224 


4741632 


12-9614814 


5 -5 17848 235 


55225 


12977875 


15-3297097! 6-171006 


169! 


2S561 


4826809 


13 0000000 


5-523775;! 236 


55696 


13144256 


15 3622915' 6-179747 


170' 


%3d00 


4913000 


13-0384048 


5-539653;: 237 


56169 


13312053 


15-3948013 


6-183463 


17l! 


2nn 


5000211 


13-0766968 


5 •550499:; 233 


56644 


13181272 


15 4272136 


6-197154 


172: 


2'J^84 


5033448 


13-1143770 


5-561293:1 239 


57121 


13651 J I'j 


15-4596248 


6-205822 


173 


29929 


5177717J 13-1529464 


5-572055 I 240 


57600 


1332401)0 


15-4919334 


6-214465 


174: 


30276 


5268024' 13-1909060 


5 53-2770:. 241 


53081 


139.)7521 


15-5211747 


6-223034 


175 


30625 


5359375 13-2237568 


5-593445! 242 


53564 


14172433 


15-5563192 


6-231630 


176; 


30976 


5451776 13-2664992 


5-604079;; 243 


59049 


14318907 


15-5384573 


6-240-251 


1771 


31329 


5545233 133041347 


5-614672!! 244 


59536 


145-26734 


15-6-204994 


6-243800 


178; 


3168i 


5639752 13-3416641 


5-625226i 245 


60025 


147061-^5 


15-6524753 


6-257325 


179i 


32041 


5735339 13-3790832 


5-635741! 246 


00516 


14836936 


15-6843371 


6-265327 


1801 


32100 


5832000 


13-4164079 


5-646216;: 247 


6100;* 


15069223 


15-7162336 


6-274305 


181! 


32761 


59-29741 


13-4536-240 


5-656653 1 248 


61504 


15252992 


15-7480157 


6-232761 


182i 


33124 


6023568 


134907376 


5-6670sIi' 249 


6-2001 


15433249 


15-7797338 


6-291195 


1831 


334S9 


6128487 


13-5277493 


5-677411; 250 


6-2500 


15625000 


15-8113333 


6-299605 


184! 


33336 


62295U4 


13-5646600 


5-637734; 251 


63001 


15313^51 


15-842)795 


6-3J7994 


185 


34225 


6331625 


136014705 


5-6930l9|; 252 


63504 


16003008 


15-8745079 


6-316360 


18r. 


34596 


6434356 


13-6331317 


5-708267 253 


64009 


16194277 


15-9059737 


6-324704 


187 


34969 


6539203 


13-6747943 


5-718479|i 254' 


64516 


16337064 


15-9373775 


6-333026 


183 


35344 


6644672 


13-7113092 


5-723654!. 255' 


650-25 


16581375 


15-9687194 


6-341326 


189; 


35721 


6751269 


13-7477271 


5-733794!! 256! 


65536 


16777216 


16-0000090 


6-349604 


190 


36100 


6859000 


13-7840433 


5-748397; 


257 


66049 


16J74593 


16-0312195 


6 357861 


191! 


364-31 


6967871 


13-8202750 


5-753965 


253 


66564 


17173312 


16-0623734 


6-366097 


19-i 


36864 


7077833' 


13-8564065 


5-763993 


259 


67031 


17373979 


16-0934769 


6-374311 


193i 


372i9 


7189057; 


13-8924 i40' 


5-778996 


260! 


67G09 


17576000 


16-1245155 


6-38-2504 


1941 


37,i36 


7301334I 


13-9283883 


5-783960 


261 


63121 


17779581 


16-1554944 


6-390676 


195; 


38025 


7414875 


13-964-2400 


5-793390 


262 


6-644 


17984723 


16-1861141 


6-398823 


1%: 


3S416 


75-29536 


14-0000000 


5-308786 


263; 


69169 


18191447 


16-2172747 


6-406958 


197: 


38809 


7645373 


14-0356683 


5-818648 


264| 


69C)96 


18399744 


16-2480768 


6-415069 


19Si 


39204 


7762392 


14-0712473 


5-323477 


265 


70225 


18609625 


16-2738206 


6-4231D8 


1991 


39601 


7880599 


14-1067360 


5-833272 


266 


70756 


18821096 


16-3095061 


6-431223 


201.' 


40000 


8000000 


14-1421356 


5 -818035 


267 


71289 


1U034163 


16-3401346 


6-439277 


201 


40401 


8120601 


14-1774469 


5-857766 


268 


71824 


19248832 


16-3707055 


6-447306 



16 



APPENDIX 



No. 


j Square. 


Cube. 


Sq. Root. 


CubeRoot. 

1 


No. 


Square. 


Cube. 


Sq Root. 


CubeRoo*. 


269 


72361 


19165109 


16-4012195 


6-455315' 


336 


1 12896 


37933055 


18-3303028 


6-952053 


270 


72900 


19633000 


16-4316767 


6-463304' 


337 


113559 


38-272753 


18-3575598 


6-958943 


271 


73441 


19902511 


16-46-20776 


6-471274 


33:5 


114^44 


33<)14472 


18-3847753 


6-965320 


272 


73984 


20123618 


16-49-24225 


6-4792-24 


339 


114921 


38953219 


18-4119526 


6-97-2683 


273 


74529 


20346417 


16-5227116 


6-437151' 


310 


11550;; 


39304000 


18-4390839 


6979532 


274 


75076 


20570824 


16-55-29454 


6-495065 


341 


110231 


39551821 


184661853 


6-936368 


275 


75625 


20796375 


16-5331240 


6-502957 


342 


116964 


40001633 


18-4932420 


6-993191 


276 


76176 


21024576 


16-6132177 


6-510330| 


343 


117649 


40353607 


18-5202592 


7 000000 


277 


75729 


21253933 


16-6433170 


6-5 135 ill 


314 


1 13336 


40707531 


18-5472370 


7-006796 


278 


77234 


21484952 


16-67333-20 


6-5-25519, 


345 


1 19025 


4105.3625 


18-5741756 


7-013579 


279 


77311 


2171763.) 


16-7032931 


6534335! 


346 


119716 


41421736 


18-6!)1()752 


7-0-20349 


280 


784(X) 


21952000 


16-7332005 


6-512133 


317 


12)409 


41781923 


18-6279360 


7027106 


281 


78961 


22183041 


167630546 


6-5499 12' 


34^ 


121104 


42144192 


18 6517531 


7-033350 


282 


79524 


2242576^ 


16-7923556 


6-557672 


349 


121801 


425 )3549 


13-6815417 


7-040531 


283 


80089 


22665187 


16 822603-^ 


6-555414' 


350 


1-22500 


42375 )00 


18-7032859 


7-047299 


284 


80656 


229063>1 


16 8522995 


6-573139 


351 


123-29! 


43243551 


13-7319940 


7-051(M)4 


235 


81225 


23149125 


16-33 !9 130 


6-5<0>j4' 


352 


123901 


43614208 


18-761663) 


7-060697 


286 


81796 


23393656 


16 9115315 


6 5SS532I 


353 


1215:)9 


43-J3-.977 


18-733-2942 


7-067377 


287 


82369 


^23639903 


16-9410743 


6 5'J5202' 


354 


125316 


44361864 


18-8148877 


7074044 


238 


82944 


23337872 


16-9705527 


6-t;;.3;54; 


355 


126025 


44733875 


18-8414437 


7-030699 


289 


83V21 


241375G9 


170000000 


6-511439 


356 


126736 


45118016 


18-8679623 


7-037341 


290 


84100 


24389000 


17-0293364 


6-519106 


357 


127449 


45199293 


18-8944436 


7-093)7] 


291 


84681 


24642171 


17-0537221 


6-626705 


353 


128164 


45382712 


18-92')8379 


7-K)0533 


292 


85264 


24897083 


17-0830075 


6-53 12 S7. 


359 


123881 


46263279 


18-9472953 


7-107194 


293 


85349 


251*^3757 


17-1172423 


6 641352 


350 


129600 


46656000 


18-9736660 


7-113737 


294 


86136 


25112184 


17-14542^2 


u-549400 


361 


130321 


470 15 -331 


19-0000000 


7-120367 


295 


87025 


25672375 


171755640 


6C)56930 


352 


131041 


47437928 


19 0262976 


7-126935 


296 


87616 


25934336 


l7-20465>)5 


6-664444; 


363 


131769 


47832147 


19-0525589 


7-133492 


297 


83209 


26198073 


17-2336379 


6-571940 


354 


132495 


43223544 


19-0737840 


7-1400.37 


293 


83304 


261G3592 


17-2626765 


6-67912) 


355 


133225 


43627125 


19-1049732 


7-146569 


299 


89401 


26730899 


17-2916165 


6-685333 


355 


133.%6 


49027395 


19-1311265 


7 15309^^ 


300 


90000 


270000CO 


17-3205031 


6-69432.) 


357 


134639 


49430863 


19-1572441 


7-15959^^ 


301 


90601 


27270901 


17-3493516 


6-701759 


363 


135424 


49835032 


lv^-1833261 


7-166095 


302 


91204 


27543603 


17-3731472 


6-70.) 173 


369 


136161 


5)243409 


19 2093727 


7-172531 


303 


91809 


27318127 


17-4063952 


6-71557^), 


370 


135900 


5065390',) 


19-2353341 


7-179054 


304 


92416 


28094464 


17-4355953 


6-723.}5!| 


371 


137641 


51054311 


19-2613803 


71855iC) 


305 


93025 


28372625 


17-4642492 


6-7313151 


372 


133334 


51473343 


19-2373)15 


7-191965 


3)6 


93636 


28652616 


17-4923557 


6-73>!654' 


373 


139129 


51895117 


19-31.32079 


7-193405 


307 


94249 


28934443 


17-5214155 


6-745997, 


374 


139376 


523136-24 


19-3390796 


7-204332 


308 


94864 


29218112 


17-5499-233 


6-753313, 


375 


1405-25 


52734375 


19-3649167 


7-2 1 1248 


309 


95481 


29533529 


17 573335 -i 


6-76:!6l4l 


376 


in 376 


53157376 


19-39' >7194 


7-217652 


310 


96100 


2979101)0 


17-6063169 


6-7'6739.7 


37: 


142129 


535S-2f,33 


19-4164878 


7-224045 


311 


96721 


3(V)30231 


17-635i;>21 


6-77515:)' 


378 


1423-^4 


54010152 


19-44-22221 


7-23)427 


312 


97344 


30371323 


17-6635217 


6-732123 


379 


143541 


54439939 


19-4679223 


7-236797 


313 


97969 


30664297 


17-6913060 


6-739661 


330 


144490 


5437-21)00 


19-4935337 


7-213156 


3U 


98596 


30959144 


17-7200451 


6-795831 


3S1 


145161 


5530C)3il 


19-519-2213 


7-249504 


315 


99225 


31255375 


17-74323J3 


6-304;);!2 


332 


145.124 


55742963 


19-54432i)3 


7-255311 


31B 


99856 


315.54496 


17-7763S-33 


6-31 12 35 


333 


1466 39 


56131.387 


19-5703353 


7-262167 


317 


100489 


31855013 


17-804493S 


6-313462 


384 


147456 


55623104 


19-5959179 


7-263432 


318 


101124 


32157432 


17-8325545 


6-8-25524 


3^5 


143225 


57066625 


19-6214169 


7-274786 


319 


101761 


32461759 


17-3605711 


6-832771 


336 


143996 


57512456 


19-6463327 


7-231079 


320 


102400 


32763000 


17-83^5t33 


6339.)04 


337 


149769 


57950603 


19 67-23156 


7-287362 


321 


103041 


33076161 


17-9164729 


6-347-)21 


338 


1.50514 


58 11 1072 


19-6977156 


7 293633 


322 


103684 


333362 J3 


17-9443534 


6 854124 


339 


151321 


53^^63859 


19-72308-29 


7-299394 


323 


104329 


33598267 


17-9722008 


6-88 !212 


390 


152100 


59319000 


19 7434177 


7-306341 


3i4 


104976 


34012224 


18-OOOOOOi) 


6-353235 


391 


15-23 SI 


59776471 


19-7737199 


7 312333 


325 


105625 


34323125 


13-0277564 


6-875344 


392 


153564 


69236233 


19-7989899 


7-318611 


326 


106276 


34645976 


13-0551701 


6-332339 


393 


154449 


60593457 


19-3-24-2-276 


7-32 48-29 


327 


106929 


34965733 


180331413 


6-339419 


394 


155236 


61162934 


19-8494332 


7-331037 


328 


107581 


35237552 


13- 1107703 


6-8.)6135 


395 


156925 


6162.)375 


19-8746069 


7-337234 


329 


108241 


35611239 


18-1333571 


6 903136 


396 


15->816 


62099135 


19-8997437 


7-3434-20 


330 


108900 


35937000 


18-1659021 


6-910423 


397 


157699 


62570773 


19-9243538 


7-349597 


331 


109561 


36264691 


18-1931054 


6-917395 


398 


158404 


63044792 


19 9499373 


7-355752 


332 


110224 


36594368 


18-2208572 


6-924355 


399 


159201 


63521199 


19-9749814 


7361918 


333 


110839 


35926037 


13-2482 S76 


6-9313)1 


400 


160000 


64UO00UO 


20-0000000 


7 3630(;3 


334 


111556 


37259704 


18-2756569 


6-933232 


1 401 


160301 


64481201 


1 20 0249844 


7-374193 


335 


112225 


37595375 


18-3030052 


6-945150 


4'.)2 


161604 


64954308 


! 20 049-J377 


7-330323 



APPENDIX. 



17 



No. 


Square. 


Cube. 


Sq. Root. 


CubeRooU 


No. 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


403 


162409 


65450827 


20-0748599 


7-386437 


470 


220900 


103323000 


21-6794834 


7-774980 


404 


163216 


65939264 


200997512 


7-392542 


471 221841 


104487111 


21-7025344 


7-780490 


405 


164025 


66430125 


20-1246118 


7-398636 


472 222784 


105154048 


21-7255610 


7-785993 


406 


164836 


66923416 


20-1494417 


7-404721 


473 223729 


105823817 


21-7485632 


7-791487 


407 


165n49 


67419143 


20-1742410 


7-410795 


474 


224676 


106496424 


21-7715411 


7-796974 


408 


166464 


67917312 


20-1990099 


7-416859 


475 


225625 


107171875 


21-7944947 


7-802454 


409 


167281 


68417929 


20-2237484 


7-422914 


476 


226576 


107850176 


21-8174242 


7-807925 


410 


168100 


68921000 


20-2484567 


7-428959 


477 


227529 


108531333 


21-8403297 


7-813389 


411 


168921 


69426531 


20-2731349 


7-434994 


478 


228484 


109215352 


21-8632111 


7-818846 


412 


169744 


69934528 


20-2977831 


7-441019 


479 


229441 


109902239 


21-8860686 


7-824294 


413 


170569 


70444997 


20-3224014 


7-447034 


480 


230400 


110592000 


21-9089023 


7-829735 


414 


171396 


70957944 


20-3469899 


7-453040 


431 


231361 


111284641 


21-9317122 


7-835169 


415 


173225 


71473375 


20-3715488 


7-459036 


482 


232324 


111980168 


21-95449S4 


7-840595 


41f. 


173056 


71991296 


20-3960781 


7-465022 


483 


233289 


112678537 


21-9772610 


7-846013 


417 


173889 


72511713 


■20-4205779 


7-470999 


484 


234256 


113379904 


22-0000000 


7-851424 


418 


174724 


73034632 


20-4450483 


7-476966 


485 


235225 


114084125 


22-02-27155 


7-856828 


419 


175561 


73560059 


20-4694895 


7-482924 


486 


236196 


114791256 


22 0454077 


7-862224 


420 


176400 


74088000 


20-4939015 


7-488872 


487 


237169 


115501303 


22-0680765 


7-867613 


421 


177241 


74618461 


20-5182845 


7-494811 


488 


238144 


116214272 


22-0907220 


7-872994 


422 


178084 


75151448 


20-5426386 


7-500741 


489 


239121 


116930169 


22-1133444 


7-878368 


423 


178929 


75686967 


20-5669638 


7-506661 


490 


240100 


117649000 


22 1359436 


7-883735 


424 


179776 


76225024 


20-5912603 


7-512571 


491 


241031 


118370771 


22-1585198 


7-889095 


425 


180625 


76765625 20-6155281 


7-518473 


492 


212064 


119095488 


22-1810730 


7-894447 


426 


181476 


77308776 20-6397674 


7-524365 


493 


243049 


119823157 


22-203d033 


7-899792 


427 


182329 


77854483 20-6639783 


7-530248 


494 


244036 


120553784 


22-225 llO:i 


7-905129 


428 


183184 


78402752 20-6881609 


7-536122 


495 


245025 


121287375 


22-2185955 


7-910460 


429 


184041 


78953589 20-7123152 


7-541987 


496 


246016 


122023936 


22-2710575 


7-915783 


430 


184900 


79507000 20-7364414 


7-547842 


497 


247009 


122763473 


22-2934968 


7-921099 


431 


185761 


80062991 20-7605395 


7-553689 


498 


248004 


123505992 


2-2-315913r) 


7-926403 


432 


1866*24 


80621568 20-7846097 


7-559526 


499 


249001 


124-251499 


22-3383079 


7-931710 


433 


187489 


81182737 20-8086520 


7-565355 


500 


250000 


125000000 


22-3606798 


7-937005 


434 


188356 


81746504 20-8326667 


7-571174 


501 


251001 


125751501 


22-383U293 


7-942293 


435 


189225 


82312875 20-8566536 


7-576985 


502 


252001 


126506008 


22-4053565 


7-947574 


436 


190096 


82881856! 20-8806130 


7-582786 


503 


253009 


127263527 


22-4276615 


7-95-2848 


437 


190969 


83453453j 20-9045450 


7-588579 


504 


254016 


128024064 


22-4499443 


7-953114 


438 


191844 


84027672 20-9284495 


7-5J4363 


505 


255025 


128787625 


22-4722051 


7-963374 


439 


192721 


84604519 20-9523268 


7-600138 


506 


256036 


129554216 


22-4944438 


7-9686-27 


440 


193600 


85184000 20-9761770 


7-605905 


507 


257049 


13032.3843 


22-51666U5 


7-973873 


441 


194481 


857661211 21-0000000 


7-611663 


508 


253064 


13109b512 


22-5388553 


7-97911-4 


4421 195364 


86350888: 21-0237960 


7-617412 


509 


259081 


131872229 


22-5610283 


7-984344 


443 196249 


86938307 21-0475652 


7-623152 


510 


260100 


132651000 


22-5331796 


7-989570 


444 


197136 


87528384i 21-0713075 


7-628884! 511 


261121 


133432831 


22-6053091 


7-994788 


445 


198025 


88121125' 21-0950231 


7-634607 


512 


262144 


134217728 


22-6274170 


8000000 


446 


198916 


887165361 21-1187121 


7 640321 


513 


263169 


135005697 


22-6495033 


8-005205 


447 


199809 


89314623 21-1423745 


7-646027 


514 


264196 


135796744 


22-67151381 


8-010403 


448 


200704 


89915392: 21-1660105 


7-651725 


515 


265225 


136590875 


22-6936114 


8-015595 


449 


201601 


90518849: 21-1896201 


7-657414 


516 


266256 


13738S096 


22-7156334 


8-020779 


450i 202500 


91125000! 21-2132034 


7-663094 


517 


267289 


13818,^413 


22-7376340 


8-025957 


451 203401 


9173385li 21-2367606 


7-6687661 


518 


268324 


138991832 


22-7596134 


8-031129 


452 204304 


92345408 21-2802916 


7-674430 


519 


269361 


139793359 


22-7815715 


8 036293 


453i 205209 


92959677 21-2837967 


7-680086 520 


270400 


140608000 


22-8035085 


8-041451 


454! 206116 


93576664; 21-3072758 


7-685733,! 521 


271441 


141420761 


22-8254-244 


8-046603 


45 3 i 207025 


94196375: 21-3307290 


7-691372 522 


272484 


142236648 


22-84731i^3 


8-051748 


4561 207936 


94818816 21-3541565 


7-697002! 523 


273529 


143055667 


22-8691933 


8 056886 


457i 208849 


95443993 21-3775583 7-702625;! 524 


274576 


143877824 


-22-8910463 


8 062018 


458 209764 


96071912 21-4009346] 7-708239i 525 


275625 


144703125 


22-9128785 


8-067143 


459 210681 


96702579 21-4242853 7-713845;! 526 


276676 


145531576 


22-9346 S99 


8-072262 


460 211600 


97336000 21-4476106 7-719443|l 527 


277729 


146363183 


22-9564806 


8-077374 


461 212521 


9797218L 21-4709106 


7-725032,; 528 


278784 


147197952 


22-9782506 


8-082480 


4621 213444 


986111-28 21-4941853 


7-730614.! 529 


279841 


148035889 


23-0000000 


8-087579 


463: 214369 


99252847, 21-5174348 


7-736188 1 530 


280900 


148877000 


23-0^1728j 


8092672 


464' 215296 


99897344 215406592 


7-741753:| 531 


281961 


149721291 


23-0134372 


8-097759 


465; 216225 


100544625; 21-5638587 


7-74731 Ifj 532 


283024 


150568768 


23-065125^ 


8102839 


466! 217156 


101194696: 21-5870331 


7-752861 li 533 


284089 


151419437 


23-0867928 


8-107913 


467 218089 


101847563 21-6101828 


7-7584021 534 


285156 


152273304 


23-1084400 


8-11-2980 


468 219024 


102503232 21-6333077 


7-763J36 535 


286225 


153130375 


23-1300670 


8 118041 


^69 219961 


103161709 21-6564078 


7-769462! 536 


287296 


153990656 


23-1516738 


8- 123098 



3# 



18 



APPENDIX. 



Xc, £:^irr. 



DC-- 

53-- 



Cai>e. 



541 



Sa. Rc-ou CubeRoo^ No. Square 



IV^D 512S145 

-^70 S•1331^r 

-35 S•13^•>i3 

01 ^•143■253 

: -1532?4 



551 

552 

553 

554 

55:' 

556 

557 

55S 

55^ 

560 

561 

562 

563 

564 

565- 

566 

569 
570 
571 
572 
573 
574 
575 
576 
57T 
57S 

580^ 
581' 
562 
583? 
554 
5-5 
5S6 
5:7 
5SS 
5S9 
^> 
591 
592 
593 
594 
595 
596( 
597; 



£001 

60l| 
£02) 



3:'3n<:>i 



31Cf24^ 
311364 
3124S1 
3136«» 
314721 
315S44 
316969 
313096 
319225 
3&I356 
3214891 

323761 

32490i;i 

326041 

327154 

32S329 

32-9476 

33!-^25 

331776 

332929 

334054 

335241 

3364i» 

337551 

3337^4 

33983») 

341056! 

3422K' 

343395 

344569 

345744 

346921 

345>1'?:) 

34>2S1 

35'>i64 

351649 

352596 

354025 

355216 

^6409 

357604 

3555501 

3600iX) 

361201 

362404 

363609 



16'^2>4151 
■ " - " "W I- 
:i?77 
■ -U^A 
~ .':3?75 
iT:>7ir*^-15 
1725*j>6?? 
1737411 1•- 
174676^7-- 
1756 16:0.' 
17655545>1 
1775 1432: 
17S453547 
17^:>t.l44 
1>03621C5 
1S1321496 
1S-22S4253 
iS325«:432 
15422j»:*.- 

iS5iio:w 

1^616^11 
1S714--245 



191102976 

1931 - 

1 ^ ' • ■- - 



1^::-' - 

19?!;;,- 
1991767^4 
200201625 

2013->>:'56 
2;;r2262>Xi3 
2<:'3297472 
2:433646.^ 
2- 537 .-<«.».' 
2»:'6425:CI 
207474f•^^ 
2C»5o275o7 
2>.'i^5S4554 
210644S75 
21170^736 
212776173 
213547192 
214921799 
216»XO»> 
21 70S 1501 
21>1672>j5' 
219256227 



/:ii 

,. . ^^_9i' 

2347S3:9-: 
234946502 
23-515952>:> 
23 5372046 
23 55543SJ 
23-5796522 
-■36>J>474 

23 622.-.>236 
23c431rOS 
23-6643191 
•-3-65543S5 
23-7<:-65392 
23-7275210 
23-745.6542 
2376^^7255 
23-7.a:7545 
23-311761^ 
23>3275j6 
23->o372^:'9 
23S746725 
23S.-5-3(«3 
23-i^ 165215 

2?;-4:>4 

- ,.71 
-0 .-7.-1576 
24-0<X)i>Xfcl' 
•24'-: -243 

- . , ; -" 2:*6 

- ~. . . ^ 1 S-S 

- . : : <9i 

-V . ::il6 
. . 1246762 
^^ 1453929 
-4 166!.»t'19 
241^7732 
i4 2.'7436y 
24-22 -*>'29 

24 2iS7ll3 
-4 26 .'3222 
2 i -25 .A- 156 
•-4-3k4-16 
243310501 
243515913 
24-3721152 
24-392521? 
244131112 
24-4335534 
24-454<:"355 
244744765 
24 494^74 
24-5153*13 
24 5356s>3 
24 55605^3 



S-i733i;-2 
S17S"2-<9 
S-lS3-:69 
S-1SS244 
5-193213 
Sr9S175 
5-203132 
S 205052 
5-213i>27 
•--- 17966 
s-22-259> 
i-2-:7525 
5-:-32746 
5-237661 
S-242571 
5-247474 
5-252371 
5257263 
S-^y2U9 
S-267029 
5 271904 
5-276773 
5-2S1635 
3-2?o493 
>-291344 



6<1^ 
61l' 
611 
612 
613 
614 
615 
616 
617 
615 

eii^ 
62i;> 

621 
622 
623 
624 



!'0.o>':o 
?-3lC»694 
-=^-315517 
S-32j3:-5 
5 325147 
5-329954 
S-334755 
5-339551 
S-344341 
5-349126 
'i-353905 
H-355<rr3 
5363447 
^ 3-35-2'>J 
5-37->r67 
5-377719 
53->'24^ 



S-4i>139S 
5-406115 
S410533 
S415542 
5-420246 
5-424945 
5•42963^ 
5-434327 
5-439..11U 
544365S 
?-44^36«> 



37urc?i 
372 liX' 
373321 
374554 
37576ij 
376996 
37<225 
379 i56 
3hOS>V 
3519-24 
3?3161 
354400 
3-5641 
3>65iM 
35-1-2V 
3?--376 



Cube, 

11 ■''4:^'?64 
, . -^25 

:-.Vil6 

22364-543 
224755712 
22556fo2.- 



23-'346397 
231475544 
2326Ct5375 
233~44-9-^ 



S^. Roc/L C^":eR-»i. 



24 57&4115 
24-5967473 
24-617C<673 
24-637 37w 
24-657656<.> 
24-6779254 
24 6951751 
24-7154142 
247335335 
24-75?.33>S 
24-779I-234 
24-7991935 
24--19?>:^ 



24.r4154- 24- r 



62> 3^4354 
629 3.-5641 
6a. ^^.-.K 

631 3--'5l61 

632 3;»9424 

633 4'Xtr.^- 

634 4jli'56 

635 4ti3225 

636 4:4,96 

637 4-576^ 
635 4'-7.44 
639 40^321 
641J 4C^?6t^J 

641 410S51 

642 412164 

643 413449 

644 414735 

645 41&-'25 

646 41731-^. 

647 415: 
643 419. 
&19 421-- : 
65 :> 422: 

651 423>..l 

652 425104 

653 4264/^ 

654 427716 

655 42A»-25 

656 43 336 

657 4316 i? 
655 43-J9r4 
659 434251 
66<i 43560?.t 

661 436921 

662 433244 

663 43^*559 

664 44-'>^ 

665 44-2225 

666 443556 

667 444559 
665 446-224 

669 447561 

670 44^901? 



2i53l437r 
24^ i- 13^3 
247573152 
245555 15i^ 
25X»470Ol> 
2512395.^1 
25J43o'9f5 
25363':' ir 



253474553 
25y694:C2 
26L'917ir.-' 



i''-"^ 



-«oS;^i-i5i 

277j673.j5 
27>445077 
27:-72-'-,'*^ 



25-Ul99i*2':> 
25-C3 •-'-" 
25 • .-, 
25 •'-^:-,. 

25-0^;;'-<'A»? 

25-1197134 
251395102 
25 1594913 
•-5-17i'3566 
■25-1992C63 
25 -2 191-404 
25 23s:5?V 
25-2536619 
-5 27>4493 
252952213 
•^5-3179773 
25-3377 1 - 
25 35744.' 

■-5 3>k: 

23-4:-5r 



5 453:e3 
5457691 
5 4--;345 

^•4.67:'X' 
5 4' 1647 
^-476•25■9 
545i:'9-26 
5455553 
5 4i»:tl55 
5-4943':>6 
5499423 
5-5:44:135 
5-5i:!3642 
5-513243 
5-51734':' 
- ' , ; - ?3 

V .9 

1 



5554437 

- 55599' ' 
■ 5''3533 
r ;63^»5l 
> 572619 
5577152 
?551651 
3-5352»:>5 
55.X»724 
5595235 
5-599743 
5-6«:4252 
5-6:i?753 
s-613243 
5-617739 
5 "6-22225 

- -•:-"* «6 

:-3 

23 




3cKj763l>'w>j 



2547547>4 5-'~. 

25-495 :'976 5-6-= ■-:?r:! 

25-5147016 5-666i31 

25-534-291.17 5-671266 

25-553>647 5 675697 

25-57ci4-::r- 3^*1104 

: ■ ; ■ : h-il 

. - 76 

^ - "34 

c.v--"-, 5t:-"'^s 

Tv'.-^v-'.S 3 710.--33 

25-7-293507 ; 715373 

2n-4-rs64 5719760 

""- .:5 3 724141 

- -.39 5-7-25513 

: -^ -.'55 5 732^2 

: ?--':;-43i 8-737260 

: -.: ■.6<.) 3-7416-25 

:>: 10343 6-745955 

2o^^43552 5 75:'34i» 



APPENDIX. 



19 



No. 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


No. 

738 


Square.! Cube. i Sq. Root. CubeRooU 


671 


450241 


302111711 


25-9036677 


8-754691 


544644 


401947272' 27-1661554 9-036886 


672 


451584 


303464448 


25-9229628 


8-759038 


739 


546121 


403583419 '27-1845544 9040965 


673 


452929 


304821217 


25-9422435 


8-763381 


740 


547600 


405224000 27-20-29410 9-0450i-i 


674 


454276 


306182024 


25-9615100 


8-767719 


741i 5490811 4U6369021 27-2213152 9-0iJiU| 


675 


455625 


307546875 


25-9807621 


8-77205311 742i 550564] 408518488 27-23967691 y-053l83| 


676 


456976 


308915776 


26 0000000 


8-776383 


743 


552049 410172407; 272580263' 9-0J7-.48 


677 


458329 


310288733 


26-0192237 


8-780708 


744 


553536; 41L>30784' 272763634; 9-06i^Ki 


678 


459684 


311665752 


26-0381331 


8-785030 


745 


5550-25i 4134^3625 -47-2946881^ 9-Oi;-53oo 


679 


461041 


313046839 


26.0576284 


8-7893471 746| 


556516j 415160936. -i7-31300o6! ^06^422 


680 


462400 


314432U00 


26-0768096 


8-793659' 1 7471 


558009; 4168327-23, 27-3313007! ^ 07 3473 


681 


463761 


315821241 


26-0959767 


8-797968! 


748 


559504 418508992^ -27-34958871 9 077520 


682 460124 


317214568 


26-1151297 8-8022721 


749 


56100r 420189749 27-3678644 sOol563 


683 


466489 


318611987 


26-1342687 8-8065721 


750 562500: 42l87.!)000: 273861279 9'085l.03 


684 


467856 


320013504 


26-1533937 


8-8108681 


751 564uOi: 4-23564751! 27-404371-2! 9-0o;.63i. 


685 


469225 


321419125 


26-1725047 


8-815160 


752| 565504 4:i5-259UU8 •27-4226184 9-Ut'o67-^ 


686 


470596 


322828856 


26-1916017 


8-819447 


753 567009J 426957777 -xi/ -4408455 i?-Ui,7V0l 


687 


471969 


324242703 


26-2106818 


8-823731 


754 568516 4-3661064' -^7-45906^4: 9-lOl7-.it) 


688 


473344 


325660672 


26-2297541 


8-8260101: 755 


5700-251 430ii68875; 27-4772633; a-iwj.-io 


689 


474721 


327082769 


26-248S095 


8-832285'| 756 


571536! 43:i08l21b 27-4'j54542 i7-io9767 


690 


476100 


328509U00 


26-2678511 


8-836556 


757 


573U49 4337980^3 •47-5i3b33-j '.;-j-i3/c2 


691 


477481 


329939371 


26-2868789 


8-840823 


758 


574564 4355195U 27-5317^^8 y-il77b3 


692 


478864 


331373888 


26-3058929 


8-845085 


759 


576081: 437-245479! •27-549'j54b ■i>-rzLi<Jl 


693 


480249 


332812557 


26-3248932 


8-84y344 


760 


57760o! 438ii7600O 27-568-J975 i>-i25d05 


694 


481636 


334255384 


26-3438797 


a -853598 


761 


57^121 440711081 -27-5802284 y-i-2^'b'.6 


695 


483025 


335702375 


26-3628527 


8-857849 


762 


580644 44-2450728 27-6043475| 


y-l33;50b 


696 


4»4416 


337153536 


26-3S18119 


8-862095 


763 


552169! 444194947 


Z7 6224546 


y-lb77..'V 


697 


485809 


338608873 


26-4007576 


8.866337 


1 764 


583696; 445943744 


2r64u549-j 


•./-i41To< 


698 


487204 


340068392 


26-41968^6 


8-870576 


765 


585-225! 447697125 


•27-658u334 


i;-i4 j77-i 


699 


488601 


341532099 


26-4386081 


8-074810 


766 


58b756j 44ij455096 


27 67o705o 


y-i4-j7.Jc 


700 


49U000 


343000000 


26-4575131 


8-879040 


767 


588;;o9j 45l2i'i663 


'47'6;^4764o 


y- 153737 


701 


491401 


344472101 


26-4764046 


8-883266 


768 


589824 45-29«483--i 


27-712812;/ 


ty-iD(7i4 


702 


492804 


345948408 


26-495-2826 


8-887488 


769 


591361] 454756609 


27-730849Z 


9-iuloo'i 


703 


494209 


347428927 


26-5141472 


8-891706 


1 770 


592900, 456533000 


27-7488739 


y-l6o65b 


704 


495616 


348913664 


26 5329983 


8-895920 771 


594441 458314011 


'27 -7668860 


y-i6Jo24 


705 


497025 


35040^625 


26-5518361 


8-9U0130 772 


5959841 4b009'.<648 


27-704^860 


y-17358.j 


706 


498436 


351895816 


26-5706605 


8-904337 773 


5975-29 461889917 


27-8028775 


y-i77544 


707 


499849 


353393243 


26-5894716 


8-908539^ 774 


59^076 


■^63684^24 


27-8408555 


y-i5ioOi^ 


708 


501264 


354894912 


26-6082694 


8-912737 


i 775 


600625 


465484375 


47-o38621o 


y-lo545c 


709 


502681 


356400829 


26-6270539 


8-916931 


776 


602176 


4o728b576 


;^7 b.5677bo 


y-l8»d0.4 


710 


504100 


357911000 


26-6458-^52 


8-921121 


777 


6037-29 


46yoy7433 


•47-8747197 


9 1^3347 


711 


505521 


359425431 


26-6645833 


8-925308 


778 


605-284 


^709ioy;)2 


-47-oy265l4 


y-iy7-~yo 


712 


506944 


360944128 


26-6833281 


8-929490 


779 


606841 


4727-.i9i3y 


•27 y 1057 15 


y -40 122b 


713 


508369 


362467097 


26-7020598 


8-933669 


780 


60840 


47455-2000 


-27-y2840ui 


9 -'205 164 


714 


509796 


363994344 


26-7207784 


8-937843 


781 


60jy61 


47637y541 


-47-y4b3772 


9-2oy09b 


715 


511225 


365525875 


S' -7394839 


b-9 12014 


782 


6115^4 


47c2ll768 


'27yo4-26-4y 


y-^i3025 


716 


512656 


367061696 


26-7581763 


8-946181 


783 


613089 


480048687 


•47-y821372 9-416950 


717 


514089 


368601813 


26-7768557 


8-950344 


784 


614656 


48189u304 


28-0000000 9 -2208 7 b 


718 


515524 


370146232 


26-7955220 


8-954503 


785 


6162-25 


483736625 


28 0l7o5l5 xi-^Z-il'^i 


719 


516961 


371694959 


26-8141754 


8-958658 


786 


6l77i:<6 


4855^7656 


28-0356yl5 y-42o7o', 


720 


518400 


373248000 


26-8328157 


8-962809 


1 78) 


61936^.: 


l«74434o3 


-^8-0535-403 9-2.3'2619 


721 


519841 


374805361 


26-8514432 


8-9669571^ 788 


620944 


48^303872 


2807 1337 ( y-2ot 52b 


722 


521284 


37636704^ 


26-8700577 


1 8-971101 


1 789 


62-2521 


491l6ii069 


'28-0a9l43Oj 9-24^435 


723 


522729 


j 377933067 


26-8886593 


; 8-975241 


: 790 


6:i41oO 


4y3o3yooo 


26 -106^386 y-44i33b 


724 


524176 


' 379503424 


26-907-2481 


: 8-979377 


1 791 


6-25681 


494913671 


28-12472-^2 y -44843-1 


725 


525625 


' 381078125 


26-9-25S240 


' 8-983509 


, 792 


627264 


496793080 


•28-l4-24a4o y-452i3i. 


726 


527076 


382657176 


26-9443872 


! 8-987637 


1 793 


62884y 


4y8677257 


28-i602557 y-456022 


727 


528529 


1 384240533 


26-9629375 


' 8-991762 


; 791 


630436 


50056618-1 


-4o-i7800D6i y-2J9ylI 


728 


529984 


; 385828352 


26-9814751 


8-995883 


795 


63-2025 


502459875 


-48-1 957444 i y-26.i797 


729 


531441 


! 387420489 


27-0000000 


9-000000 


1 796 


633616 


50435O336 


2«-213472o| 9-267680 


730 


532900 


i 389017000 


27-0185122 


1 9-004113 


j 79/ 


63520b 


506--i61573 


28-231 lo84i 9-471559 


731 


534361 


390617891 


27-0370117 


i 9-008223 


j 798 


6368«-M 


508169592 


48-2488y3oi 9-275435 


732 


535824: 392223168 


27-0554985 


9-012329 


; 799 


638401 


51008-4399 


28-4665081! 9-479308 


733 


1 527289| 393832837 


27-0739727 


9-016431 


1 800 


640000 


512000000 


28-28427 r^l 9-203178 


734 


538756' 395446904 


27-0924344 


9-020529 


, 801 


641601 


5139-2-2401 


28-b0ly434 


9-207044 


735 


540225' 397065375 


27.1108834 


: 9-01^624 


802 


643204 


51584y608 


28-3l9c040 


y-290907 


736 


541G96! 39S688256 


27-1293199 


9-028715 


1 803 


644809 


517781627 


28-3372540 


9-29476i 


737 


543169J 400315553 


27-1477439 


1 9-032802 


1 804 


646416 


519718464 


28-3548938 


9-298624 



20 



APPENDIX. 



No. 

b05 

806 

ti07 

808 

80'j| 

rjUH 

bill 

6U\ 

6U\ 

814' 

81:')! 

bi; 

818 
81'J: 
8-i0| 
821; 
8221 
823, 
8241 
825' 
826 
o27 
828! 
82<J 
8301 

83 1: 

832' 

833 1 
83 4 1 
835 
836! 

8371 
838i 

83yi 

840' 
8411 
8421 
843 

844 
845' 
846 

847 i 
8481 
8491 
850, 
851 
852' 
853i 
8541 
855 
856 
857 
8581 
859 1 

860; 

861 
852 
8631 
864 
865 
866 
867 
868 
869 
870 
871 



Square. 



Cube. 



Sq. Root. 



648025 521660125; 
649636 52360C616! 
651249 525557943: 
652864 527514112, 
654481 529475129i 
656100 5314410001 
657721 533411731! 
65<;344 535387328 
660969 537367797' 
662596 539353144! 
664225 541343375: 
6.i5856 543338496 
6r,74H9 54533S513 
669124 547313432' 
670761 549353259i 
672400 5513680001 
674U41 55338766 Ij 
675684 5554122481 
67732y 557441 767 i 
678976 559476224| 
680625 5615156251 
682276 563559976: 
6:)3929 565609283! 
685584 567663552 
687241 569722789 
6889(0 571787000 
G90561 573856191 
69-224 575930368 
693889 578009537 
6'.*5556 580093704 
697225 582182875 
698896 584277056 
700569 586376253 
702244 5884804721 
703921 590589719! 
705600 592704000| 
707281 594823321! 
708i^64 596947688! 
710649 599077107] 
7123361 601211534! 
714025! 6U335 11251 
7157161 605495736 
717409! 607645423 



CubeRoot. 



719104 

7j''1801 



n.'^K) 614125000 



609800192 
611960049 



724201 
725904 
727609 
729316 
7310^5 
732736 
734449 
736164 
737881 
739600 
741321 
743044 
744769 
746496 
748225 
749956 
751689 
753424 
755161 
756900 
758641 



616295051 

618470208; 

620650477 

622835864 

625026375 

6272220161 

629422793' 

631628712! 

633839779! 

636056000! 

638277381! 

640503928! 

642735647! 

614972544 

647214625! 

649461896^ 

6517143631 

653972032| 

656234909! 

658503000' 

660776311! 



28 3725219 

28 3901391 

28-4077454 

28 4253408 

28-4429253 

28-4604989 

28-4780617 

28-4956137 

285131549 

28-5306852 

28-5482048 

28-5657137 

28-5832119 

28-6006993 

28-6181760 

28-6356421 

28-6530976 

28-6705124 

28-6879766 

28-7054002 

28-72-28132 

28-7402157 

28-7576077 

28-7749891 

28-79236011 

28-8097206 

28-8270706' 

28-8144102 

28-8617394 

28-8790582 

2S-8963666 

28-9136646 

28-9309523 

28-9482297 

28-9654967 

28-9827535 

29O0000O0! 

29-0172363! 

29-0344623 

29-0511)781' 

29-0688837 

290860791 

29-1032644 

29-1204396 

29-1376046 

29- 1547595 

29-1719043 

29 1890390 

29-2061637! 

29-2232784 

29-2403830! 

29-2574777 

29-2745623 

29 2916370 

29-3087018 

29 -3257566 

29-3428015 

29-3598365 

29-3768616 

29-3938769 

29-4108823 

2^-4278779 

29-4448637 

29-4618397 

29-4788059 

29-4957624 

29-5127091 



No. 



Square. 



9-302477! 
9-306328' 
9-310175 
9-314019 

9-317860' 
9-321697; 
9-325532 
9-329363 
9-333192 
9-337017 
9-340839 
9-344657 
9-348473 
9-3522^8:! 
9-356095 1 
9-359902 I 
9-363705' 
9-367505! 
9-371302:1 
9 3750961 
9-378887i| 
9-38-2675!! 
9-3S6460 
9-390242 
9-394021 
9-3.-7796! 
9-401569! 
9-4053391 
9-4091051 
9-412869: 
9 -4166301 
9-420.387 
9-4241421 
9 -427894; 
9-431642' 
9-4353881 
9-439131; 
9-442870 
9 •446607! 
9-450341 
9-454072; 
9-457800! 
9-461525! 
9-465247; 
9-468966 
9-472682! 
9-476396! 
9-480106' 
9-483814 
9-487518. 
9-491220 
9-494919 
9-498615 
9-5023U8: 
9-505998 
9-509685 
9-513370 
9-517051 
9-5-20730 
9-524406 
9-528079 
9-531750 
9-535417 
9-539082 
9-542744 
, 9-546403 
' 9-550059 



872 

873; 

874 

875! 

876 

877 

878 

879 

880 

881 

882! 

883: 

S-M 

8b5 

886 

887 

888 

889 

890 

891 

892 

893 

894 

895 

896 

897 

898 

899 

900 

901 

902 

903 

904 

905 

906 

907 

908 

909 

910 

911 

912 

913 

914 

915 

916 

91 

918 

919 

92u 

921 

922 

923 

924 

925 

926 

92 

928 

929 

930 

931 

932 

933 

934 

935 

936 

937 

938 



Cube. 



760384 

762129 

763376 

765625 

767376 

769129 

770884 

772641 

774400 

776161 

777924 

779689 

78145(: 

783225 

78499^: 

786769 

788544 

790321 

7'j2100 

793881 

795664 

797449 

799236 

801025 

802816 

804609 

806404 

808201 

810000 

811801 

813604 

815409 

817216 

819025 

820836 

822649 

824464 

826281 

828100 

8-29921 

831744 

833569 

835396 

837225 

839056 

840889 

842724 

844561 

846400 

818-241 

850084 

851929 

853776 

855625 

857476 

859329 

8611841 

863041 

864900' 

8667611 

868624. 

8704891 

872356! 

874225 

876096 

8779691 

8798441 



Sq. Root. CubeRoot, 



663054848 

665338617 

667627624 

669921875 

tw2-221376 

674526133 

676836152 

679151439 

681472000 

683797841 

686128968 

688465387 

690807104 

693154125 

695506456 

697864103 

700227072 

702595369 

704969000 

707347971 

709732288 

712121957 

714516984 

716917375 

719323136 

721734-273 

724150792 

726572699 

729000000 

73143-2701 

7338708081 

736314327 

738763264 

741217625 

743677416 

746142643 

748613312 

751089429 

753571000 

756058031 

758550528 

761048497 

763551944 

766060875 

768575296 

771095213 

773620632 

776151559 

778688000 

781229961 

783777448 

786330467 

788889024 

791453125 

794022776 

796597983 

799178752 

801765089 

804357000 

806954491 

809557568 

812166237 

814780504 

817400375 

820025356 

822656953 

825293672 



29-5296461 

29-5465734 

29-5634910 

29-5803989 

29-5972972 

29-6141858 

29-6310648 

29-6479342 

29-6647939 

29-6816442 

29.6984848, 

29-7153159! 

29-73213751 

29-7489496 

29-7657521 

29-78254521 

29-7993289 

29-8161030 

29-83-28678 

29-8496231 

29-8663690 

29-8831056 

29-8998328 

29-9165506 

29-933-2591 

29-9499583 

29-9666481 

29-9833^7 

30-0000000 

30-0166620 

30-0333148 

30-0499584 

30-0665928 

30-0832179 

30-0998369 

30-1164407 

30-1330383 

30-1496269 

30-1662063 

30-18-27765 

30-1993377 

30-2158899 

30-2324329 

30-2489669 

30-2654919 

30-2820079 

30-2985148 

30-3150128 

30-3315018 

30-3479818 

30-3644529 

30-3809151 

30-3973683 

30-4138127 

30-4302481 

30-4466747 

30-4630924 

30-4795013 

30-4959014 

30-512-2926 

30-5286750 

30-5450487 

30-5614136 

30-5777697 

30-5941171 

30-6104557 

30-6267857 



9-553712 
9-557363 
9-561011 
9-564656 

9-568298 

9-571938 

9-575574 

9-579208 

9;582840 

9-:86468 

9-590094 

9-593717 

9-597337 

9-600955 

9-604570 

9-608182 

9-611791 

9-615398 

9-619002 

9-622603 

9-626-202 

9-62979 

9-633391 

9-636981 

9-64056^ 

9-644154 

9-647737 

9-651317 

9-654894 

9-658468 

9-66-2040 

9-665610 

9-669176 

9-672740 

9-676302 

9-679860 

9-683417 

9-686970 

9-690521 

9-694069 

9-697615 

9-701158 

9-701699 

9-708237 

9-711772 

9-715305 

9-718835 

9-7-22363 

9-725888 

9-7-29411 

9-732931 

9-736448 

9-739963 

9-743476 

9-746986 

9-750493 

9-753998 

9-757500 

9-761000 

9-764497 

9-767992 

9-771484 

9-774974 

9-778462 

9-781947 

9'7854 

9-788909 



APPENDIX. 



21 



No. 


Square.; Cube. 


Sq, Root. 


CubeRoot. 

! 


No. 

970 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


939 


881721; 827936019 


30-6431069 


9-792386 


940900 


912673000 


31-1448230 


9-898983 


910 


883600 830584000 


30-6594194 


9-795861; 


971 


942841 


915498611 


31-1608729 


9-902333 


941 


885481 


833237621 


30-6757233 


9-799334 


972 


944784 


918330048 


31-1769145 


9-905782 


942 


887364 


835896888 


30-6920185 


9-802804 


973 


946729 


921167317 


31-1929479 


9-90917"^ 


943 


889249 


838561807 


30-7083051 


9-806271 


974 


948676 


924010424 


31-2089731 


9-912571 


944 


891136 


841232384 


30-7245830 


9-809736 


975 


950625 


926859375 


31-2249900 


9 -915962 


945 


893025 


843908625 


30-7408523 


9-813199 


976 


952576 


929714176 


31-2409987 


9-919351 


946 


894916 


846590536 


3U-;571130 


9-816659 


977 


954529 


93-2574833 


31-2569992 


9-9-2273 -. 


947 


896809 


849278123 


30-7733651 


9-820117 


978 


956484 


935441352 


31-2729915 


9-926122 


94b 


898704 


851971392 


30-7896086 


9-823572 


979 


958441 


938313739 


31-2889757 


9-929504 


949 


900601 


854670349 


30-8058436 


9-827025 


980 


960400 


941192000 


31-3049517 


9-932834 


yso 


902500 


857375000 


30-8220700 


9-830476 


981 


962361 


944076141 


313-209195 


9-936-261 


951 


904401 


860085351 


30-8382879 


9-833924 


982 


964324 


946966168 


31-3368792 


9-939636 


952 


906304 


862801408 


30-8544972 


9-837369 


983 


966289 


949862087 


31-35-28308 


9-9430'J9 


953 


908209 


865523177 


30-8706981 


9-840813 


984 


968256 


952763904 


31-3687743 


9-916380 


954 


910116 


868250664 


30-8868904 


9-844254 


985 


970225 


955671625 


31-3847097 


9-949748 


955 


912025 


870983875 


30-9030743 


9-847692 


986 


972196 


958585256 


31-4006369 


9-953114 


956 


913936 


873722816 


30-9192497 


9-851128 


987 


974169 


961504803 


31-4165561 


9-956477 


957 


915849 


876467493 


30-9354166 


9-854562 


288 


976144 


964430272 


31-4324673 


9-95983U 


958 


917764 


879217912 


30-9515751 


9-857993 


989 


978121 


967361669 


31-4483704 


9-963 198 


959 


919681 


881974079 


30-9677251 


9-861422 


990 


980100 


970299000 


31-464-2654 


9-966555 


-A)0 921S00 


884736000 


30-9838668 


9-864848 


991 


982081 


973242271 


31-4801525 


9-9699.!.' 


961 


923521 


887503681 


31-0000000 


9 •868-272 


992 


984064 


976191488 


31-4960315 


9-973i6.: 


962 


925444 


S90277128 


31-0161248 


9-871694 


993 


986049 


979146657 


31-5119025 


9-976612 


963 


927369 


893056347 


31-0322413 


9-875113 


994 


988036 


982107784 


31-5277655 


9-9799G0 


964 


929296 


8958413441 31-0483494 


9-878530 


995 


990025 


985074875 


31-5436206 


9-933305 


965 


931225 


898632125 


31-0644491 


9-881945 


996 


992016 


988047936 


31-5594677 


9-98664y 


966 


933156 


901428696 


310805405 


9-885357; 


997 


994009 


991026973 


31-5753068 


9-98999(. 


967 


935089 


904231063 


31-0966236 


9-8887671 


998 


996004 


994011992 


31-5911380 


9-993;.-2.. 


968 


937024 


907039232 


31-1126984 


9-892175 


999 


998001 


99700-2999 


31-6069613 


9-996666 


969 


938961 


909853209 


31-1287648 


9-895580 


1000 


1000000 


1000000000 


31-6227766 


10-OOUOOU 



The following rules are for finding the squares, cubes and roots, of 
numbers exceeding 1,000. 

To find the square of any number divisible without a remainder. 
Rule. — Divide the given number by such a number, from the forego- 
ing table, as will divide it without a remainder ; then the square of the 
quotient, multiplied by the square of the number found in the table, 
will give the answer. 

Example- -What is the square of 2,000 ? 2,000, divided by 1,000, 
a number lound in the table, gives a quotient of 2, the square of which 
is 4, and the square of 1,000 is 1,000,000, therefore : 
4 X 1,000,000 = 4,000,000 : the Ans. 

Another example. — What is the square of 1,230? 1,230, being di- 
vided by 123, the quotient will bo 10, the square of which is 100, and 
the square of 123 is 15,129, therefore : 

100 X 15,129-= 1,512,900: the Ans. 

To find the square of any number not divisible without a remainder. 
Rule. — Add together the squares of such two adjoining numbers, from 
the table, as shall together equal the given number, and multiply the 
sum by 2 ; then this product, less 1, v*'ill be the answer. 

Example. — What is the square of 1,487 ? The adjoining numbers, 
743 and 744, added together, equal the given number, 1,487, and the 
square of 743 = 552,049, the square of 744 = 553,536, and these 
added, = 1,105,585, therefore : 

1,105,585 X 2 =- 2,211,170— i = 2,211,169: the Ans. 

To find the cube of any number aiv/sible without a remainder. 
Rule. — Divide the given number by sucn a number, from the forego- 



22 APPENDIX. 

ing table, as will divide it without a remainder ; then, the cube of the 
quotient, multiplied by the cube of the number found in the table, will 
give the answer. 

Exam.pl e,--Whsit is the cube of 2,700 ? 2,700, being divided by 900, 
the quotient is 3, the cube of which is 27, and the cube of 900 is 
729,000,000, therefore : 

27 X 729,000,000 -= 19,683,000,000: the Ans. 

To find the square or cube root of numbers higher than is found in the 
table. Mule. — Select, in the column of squares or cubes, as the case 
may require, that number which is nearest the given number ; then 
t!ie answer, when decimals are not of importance, will be found di- 
rectly opposite in the column of numbers. 

Example. — What is the square-root of 87,620 ? In the column of 
squares, 87,616 is nearest to the given number ; therefore, 296, im- 
mediately opposite in the column of numbers, is the answer, eearly. 

Another example. — What is the cube-root of 110,591 ? In the co- 
lumn of cubes. 110,592 is found to be nearest to the given number ; 
therefore, 48, the number opposite, is the answer, nearly. 

To find the cube-root more accurately. Rule. — Select, from the co- 
lumn of cubes, that number which is nearest the given number, and 
add twice tJie number so selected to the given number ; also, add twice 
the given number to the number selected from the table. Then, as 
the former product is to the latter, so is the root of the numbei' selected 
to the root of the number given. 

Example. — What is the cube-root of 9,200 ? The nearest number 
in th-e column of cubes is 9,261, the root of which is 21, therefore : 
9261 9200 

2 2 



18522 18400 
9200 9261 



As 27,722 is to 27,661, so is 21 to 20-953 4- the Ans. 
21 



27661 
55322 



27722)580881(20-953 + 
55444 



264410 
249498 



149120 
138610 

105100 
3316rt 

21934 



APPENDIX. 23 

To find the square or cube root of a whole numher with decimals. 
Rule. — Subtract the root of the whole number from the root of the next 
higher number, and multiply the remainder by the given decimal ; 
then the product, added to the root of the given whole number, will 
give the answer correctly to three places of decimals in the square- 
root, and to seven in the cube-root. 

Example. — What is the square-root of 11'14? The square-root of 
11 is 3-3166, and the square-root of the next higher number, 12, is 
3-4641, therefore : 

3-4641 

3-3166 



•1475 
•14 


the Ans. 


(See page 32. 




5900 
1475 




•020650 
3-3166 




3-33725 : 


App, 



ctULES FOR THE REDUCTION OF DECIMALS. 

To reduce a fraction to its equivalent decimal. Rule. — Divide the 
numerator by the denominator, annexing cyphers as required. 

Example. — What is the decimal of a foot equivalent to 3 inches ? 
3 inches is -^^ of a foot, therefore : 
/j ... 12) 3-00 

•25 Ans. 
Another example. — What is the equivalent decimal of f of an inch ? 
^ .... 8) 7^000 



•875 Ans. 

To Tpd"ce a compound fraction to its equivalent decimal. Rule. — In 
dccoraance with the preceding rule, reduce each fraction, commen- 
cmg at the lowest, to the decimal of the next higher denomination, to 
which add the numerator of the next higher fraction, and reduce the 
sum to the decimal of the next higher denomination, and so proceed to 
the last ; and the final product will be the answer. 

Example. — What is the decimal of a foot equivalent to 5 inches, | 



and j\ of an inch ? 



The fractions in this case are, ^ of an eighth, f of an inch, and -^ 
of a foot, therefore : 



el 



APPEND 


IX. 


2) 1-0 

•5 
3- 

8) 3-5000 


eighths. 


•4375 
5- 


inches. 



-i- 12) 5-437500 

•453125 Ans. 
The process may be condensed, thus ; write the numerators of the 
given fractions, from the least to the greatest, under each other, and 
place each denominator to the left of its numerator, thus : 



i 2 



f 8 



-a- 12 

12 



1-0 



3-5000 



5-437500 



•453125 Ans. 

To reduce a decimal to its equivalent in terms of lower denominations. 
Rule. — Multiply the given decimal by the number of parts in the next 
less denomination, and point off from the product as many figures at 
the right hand, as there are in the given decimal ; then multiply the 
figures pointed off, by the number of parts in the next lower denomina- 
tion, and point off as before, and so proceed to the end ; th<»n ^h-' -eve- 
ral figures pointed off at the left will be the answer. 

Example. — What is the expression in inches of 0-390625 feet '? 
Feet 0-3906-25 

12 inches in a foot. 



Inches 4-687500 

8 eighths in an inch. 



Eighths 5-5000 

2 sixteenths in an eighth 

Sixteenth 1-0 

Ans., 4 inches f and -V- 
Another example. — What is the expression, in fractions of ar inch, 
©f 0-6875 inches ? 



Inches 0-6875 



8 eighths in an inch. 



Eighths 5-5000 

2 sixteenths in an eighth. 

Sixteenth 1-0 

Ans., I and j\. 



TABLE OF CIRCLES. 

(From Gregory's Mathematics.) 

From this table may be found by inspection the area or circumfe- 
rence of a circle of any diameter, and the side of a square equal to the 
area of any given circle from 1 to 100 inches, feet, yards, miles, &c. 
If the given diameter is in inches, the area, circumference, &c., set 
opposite, will be inches ; if in feet, then feet, 6zc. 









Side of 








Side oi" 


Diam. 


Area. 


Circum. 


equal sq. 


Diam. 


Area. 


Circum. 


equal sq. 


^ 


•04908 


•78539 


•22155 


^ 


90-76257 


33-77212 


9-526J3 


•5 


•19635 


1-57079 


-44311 


11- 


95-03317 


34-55751 


9-74SI.; 


•75 


•44178 


2-25619 


•66467 


•25 


99-40195 


35-34291 


9-'.i700o 


1- 


•78539 


314159 


•88622 


•5 


103-86890 


3;r 1-2331 


10-1916 ; 


•25 


122718 


3-92699 


1-10778 


•75 


108-43403 


36-91371 


10-41316 


•5 


1-76714 


4-71-238 


1-32934 


12- 


1130'.)733 


37-69911 


i;;-63472 


•75 


2-40528 


5-49778 


1-5508.) 


•25 


117-85SS1 


33-48451 


10 85827 


2- 


314159 


6-28318 


1-77-245 


-5 


122-71816 


39-26990 


11-077-^-^ 


•25 


3-97607 


7-06858 


1-99401 


•75 


1-27-67628 


40 05530 


ll-2..'-..3.; 


•5 


4-90873 


7-8539S 


2-21556 


13- 


1.32-7322S 


40-84070 


li -5 j( ;'.).'): 


•75 


5-93957 


8-63937 


2-43712 


•25 


137-83646 


41 -6-2610 


ll-7i:;5- 


3^ 


7-06853 


9-42477 


2-65SGS 


-5 


1431338] 


42-4 11 50 


ii-'jr.4 its 


•25 


8-29576 


10-21017 


2-88023 


•75 


148-43934 


43-19689 


12-Ir.5;i- 


•5 


9-62112 


10-99557 


3-10179 


14- 


153-93804 


43-93-2-29 


12-407 17! 


•75 


11-04466 


11-78097 


3-32335 


•25 


159-43491 


44-76769 


12-r>2sT3! 


4^ 


12-56637 


12-56637 


3-54490 


•5 


165- 1-2996 


45-55309 


12-350-i'jJ 


•25 


14-18625 


13-35176 


3-766461 


•75 


170-87318 


46-33-S49 


13-07 l^;i 


•5 


15-90431 


14-13716 


3-98802| 


15- 


176-71458 


47-12333 


13-2V'34j 


•75 


17-7-2054 


14-92256 


4-209571 


•25 


182-65416 


47-U092S 


13-5I^-.J;i 


5- 


19-63195 


15-70796 


4-43113; 


■f) 


188-69190 


43-69463 


13-73o5i 


•25 


21-64753 


16-49336 


4-65269; 


-75 


191-8-2783 


49-48003 


13-'J5S..'7 


•5 


23-75829 


17-27875 


4-87424 


16- 


201-06192 


50-26548 


14-17y63 


•75 


25-96722 


18-06415 


5-09580J 


•25 


207-394-20 


51-05088 


14-401 Is 


6- 


28-27433 


18-84955 


5-31736 


•5 


213-82464 


51-83G-27 


14-62-27i 


•25 


30-67961 


19-63495 


5-53891 


•75 


220-35327 


5262167 


14-S443,) 


•5 


38-18307 


20-42035 


5-76047 


17^ 


226-98006 


53-40707 


15-0t'5-i5 


•75 


35-78470 


21-20575 


5-982031 


•25 


233-70504 


54-19-247 


15--287 4i 


7- 


38-48456 


21-99114 


6-20358 


•5 


240-5-28181 


54-97737 


15-5'J897 


•25 


41-28249 


2!?'7765i 


6-42514 


•75 


•247-44950 


5576326 


15-73-.;-52 


•5 


44-17864 


23-56194 


6-64670 


13^ 


264-46900 


56-54366 


15-95-20S 


•75 


47-17297 


24-31731 


6-86S25 


-25 


266 -536671 


57-33406 


16-17364 


8^ 


50-26548 


25-13274 


7-08981 


•5 


268-80252 


53-11946 


1639519 


•25 


53-45616 


25-91813 


7-31137 


•75 


276-116541 


53-90436 


1()-61()75 


•5 


56-74501 


26-70353 


7-53292 


19^ 


283-528731 


59-69026 


16-83331 


•75 


60-13204 


27-48893 


7-7544S 


•25 


291-03910! 


60-47565 


17-05936 


9^ 


63-61725 


28 27433 


7-97604 


-5 


298-64765! 


61-26105 


17-28142 


•25 


67-20063 


29-05973 


8-19759 


•75 


306-35437 


62-04645 


17-50298 


•5 


70-88218 


29-84513 


8-41915 


20^ 


314 159-26 


62-83185 


17-72453 


•75 


74-66191 


30-63052 


8-64071 


•25 


322-06233 


63-61725 


17-94609 


10- 


78-53981 


31-41592 


8-86226 


•5 1 


330 06357 


64-40264 


18-16765 


•25 


82-51589 


32-20132 


9-08382 


•75 1 


338-16-299 


65-18804 


18-38920 


5 


86-59014 


32-98672 


9-30538 


21- i 


346-36059 


65-97344| 


18-61076 



26 



APPENDIX, 









Side of ' 








Side of 1 


Diam. 


Area. 


Circum. 


equal sq. 


Diam. 


Area. 


Circum. 


equal sq. | 


Tl^ 


354-65f)35 


66^75884 , 


18-83232 


38- 


1134-11494 


119-38052 


33-67662 


•5 


363-05030 


67^54424 i 


19-05387 1 


•25 


1149-08660 


120-16591 


33-89817 


•75 


371-54241 


68-32964 


19^27543 


•5 


1164-15642 


120-95131 


34-11973 


22- 


380-13271 


69-115031 


19^49699 


•75 


1179.32442 


121-73671 


34 34129 


•25 


388-82117 


69-900431 


19-71854 


39- 


1194-59060 


122-52211 


34-56*285 


•5 


3 J7 -60782 


70-68583' 


19-94010 


•25 


1209-95495 


123-30751 


34-78440 


•75 


406-49263 


71-47r23 


20-16166 


•5 


h225 41748 


124-09290 


35(10596 


23- 


415-475fi2 


72-25663; 


20-38321- 


•75 


1240-97818 


124-87830 


35-22752 


•25 


424-55679 


7304202 


20-60477 


40- 


r25f;-63704 


125-66370 


35-44907 


•5 


433-73613 


73-82742 


20-82633 


•25 


1272-39411 


126-44910 


35-67063 


•75 


443-01365 


74-61282^ 


21 04788 


-5 


1-288-21933 


127-23450 


35-89219 


21- 


452-38934 


75-398"22 


21-26944 


•75 


1304-20273 


128-01990 


36-11374 


-25 


461-8!")3-20 


76-18362 


21-49100 


41- 


13-20-25431 


128-80529 


36-33530 


•5 


471-43524 


76-96902 


2171255 


-25 


1336-40406 


129-59069 


36-55686 


•75 


481-10546 


77-75441 


21-93411 


•5 


1352-65198 


130-37609 


36-77841 


25- 


490-87385 


78-53J81i 


22-15567 


•75 


1368-99808 


131-16149 


.36-999<)7 


•25 


500-74041 


79-32521! 


22-377-22 


42^ 


1385-44*2.36 


131-94689 


37-22153 


•5 


510^705 15 


80-11061 


22-59878 


•25 


1401-98480 


132-73-2-28 


37-44308 


•75 


520-76^06 


80-89601: 


22-82034 


•5 


1418-62543 


133-51768 


37-66464 


•.6^ 


530-9-2915 


81-68140 


•23-04190 


•7fS 


1435-3&423 


134-30308 


37-88620 


••25 


541-58842 


82-46680 


23-26345 


43- 


1452-20120 


135-08348 


3810775 


•5 


551-51586 


^ 83--25220 


23-48501 


•25 


146.)-13635 


135-87388 


38-3-2931 


•75 


562-00147 


84-03760 


23-70657 


-5 


1486-16967 


136-65928 


38-5508"*/ 


27- 


572-55526 


84-82300. 


23-92812 


•75 


1503-30117 


137-44467 


33-77-242 


•25 


583-20722 


85-60839' 


24- 14968 


44- 


1520-53084 


138-23007 


38-99398 


•5 


5'j3-95736 


86-39379 1 


•24-371'24 


-25 


1537-85869 


139-01547 


39-21554 


•75 


604-80567 


87-17919 


24-59279 


•5 


1556-28471 


139-80087 


39-43709 


28- 


615-75216 


87-96459. 


24-81435 


•75 


1572-80890 


140-58627 


39 65865 


•25 


626-79682 


88-74999; 


25-03591 


45- 


1590-43128 


141-37166 


39-88021 


•5 


637-93965 


89-53539 


25-25746 


•25 


1608- 15182 


142-15705 


4010176 


•75 


649^ 18066 


90-32078; 


25-47902 


•5 


1625-97054 


142-94246 


40-32332 


29- 


660 51985 


91-10613 


25-70058 


•75 


1643 88744 


143-72786 


40-54488 


•25 


671-95721 


91-89158' 


25-92213 


46- 


1661-90251 


144-513-26 


40-76643 


•5 


683-49275 


92-67698 


•26-14369 


•25 


1680-01575 


145-29866 


40-98799 


•75 


6 J5- 12646 


93-4623-^ 


26-365-25 


•5 


1698-22717 


146-08405 


41-20955 


3J- 


706-85834 


94-24777. 


26 58680 


•75 


1716-53677 


146-86945 


41-43110 


•25 


718 68840 


95-03317 


26-80836 


47- 


1734-9445-1 


147-65485 


41-65266 


•5 


730-61664 


95-81857 


27-02992 


-25 


1753-45048 


148-44025 


41-87422 


•75 


742 64305 


96-60397 


27-25147 


-5 


1772-05460 


149-22565 


42-09577 


31- 


75r76763 


97-38937 


27-47303 


-75 


1790-75689 


150-01104 


42-31733 


•25 


766-99039 


98-17477 


27-69459 


48- 


1809 55736 


150-79644 


42-53889 


•5 


779-31132 


98 96016 


27-91614 


-25 


1828 -4 560 J 


151-58184 


42-76044 


•75 


791-73043 


99-74556 


28-13770 


•5 


1847-45282 


152-36724 


42-98200 


32- 


804-24771 


100-53096 


28 •:! 59-26 


•75 


1866-54782 


153-15-264 


43-20356 


•'25 


816^.><6317 


101-31636 


2^. ■58081 


49^ 


1885-74099 


153-93804 


43-42511 


•5 


8-29-57681 


102^ 10176 


2Sh0237 


•25 


1J05-83233 


154-72343 


43-64667 


•75 


812-38861 


102-88715 


29 02393 


-5 


19-24-42184 


155-50883 


43-86823 


33- 


855-29859 


103-67255 


29-21548' 


•75 


1943-90954 


156-29423 


44-08978 


•25 


868-30675 


104-45795 


29-46704 


50- 


1963-49540 


157-07963 


44-31134 


•5 


881-41308 


105-21335 


29-68860; 


•25 


1983-17944 


157-96503 


44-53290 


•75 


894-61759 


106-02875 


29-910l5j 


•5 


2002-96166 


158-65042 


44-75445 


31- 


907-92027 


106-81415 


30-13171! 


•75 


2022-84205 


159-43582 


44-97601 


25 


921-32113 


107-59954 


30-353-27; 


51- 


-2042-82062 


160-22122 


45-19757 


•5 


93i-8^2016 


108-3S494 


30-574821 


-25 


2062-89736 


161-0(662 


45-41912 


•75 


948 41736 


109-17034 


30-79638' 


1 '5 


2083-07227 


161-79-202 


45-64068 


35^ 


962- J 1275 


109-95574 


31-01794' 


' -75 


2103-3 1536 


16-2-57741 


45-86-2-24 


•25 


975-90630 


110-74114 


31-23949; 


52- 


21-23-71663 


163-36281 


46-08380 


•5 


989-79803 


111-52653 


31-46105! 


•25 


2144-18607 


164-14821 


46-30535 


•75 


1003-78794 


112-31193 


31-68261; 


-5 


2164-75368 


184-93361 


46 5-2691 


35^ 


1017-87601 


11309733 


31-90416 


•75 


2185-41947 


165^71901 


46-74847 


•25 


1032-06'2-27 


113-88-273 


32- 1-2572 j 


53- 


2-206-18344 


166^50441 


46-97002 


•5 


1046-34670 


114-66813 


32-347281 


•25 


2-2-27-04557 


167-28980 


47-19158 


•75 


1060-72930 


115-4^353 


32-56883; 


•5 


-2248-00589 


168-07520 


47-41314 


37^ 


1075-2100S 


116-23892 


32-79039 


•75 


2269-06438 


168-86060 


47-63469 


•25 


1089-78903 


117^02432 


33-01195 


54- 


j -^290-22104 


169-64600 


47-85()-25 


•5 


1104-46616 


117-80972 


33-23350 


•25 


1 2311-475-^8 


170-43140 


48-07781 


•75 


11 19^24147 


1 18-59572 


33-45506 


i -5 


2332-8-2889 


171-21679 


48-29936 



APPENDIX. 



27 



1 — 






Side of 


1 






Side of 


iDiam. 


Area. 


Circura. 


equal sq. 


1 Diam. 


Area. 


Circum. 


equal sq. 


"5475 


2354-28CH)3 


17200219 


48-52092 


71-5 


4015-15176 


2-24-62337 


63-365-22 


55- 


2375-82944 


172-78759 


48-74248 


•75 


4043-27833 


2-25-40927 


63-53678 


•25 


2397-47698 


173-57-299 


48-96403 


72- 


4071-50407 


226-19467 


63-80833 


•5 


2419-22269 


1 174-35839 


49-18559 


•25 


4099-82750 


226-98006 


64-0-2989 


•75 


2441-06657 


175-14379 


49-40715 


•5 


4123-24909 


227-76546 


64-25145 


55- 


246300864 


175-9-2918 


49-62870 


•75 


4156-76386 


2-28-55086 


64-47300 


•25 


2185-04887 


176-71458 


49-85026 


73- 


4185-33681 


229-336-26 


64-69456 


•5 


2507- 187-28 


177-49998 


50-07182 


•25 


4214-10-293 


230-12166 


64-91612 


•75 


25-20-42387 


178-28538 


50-29337 


■5 


4242-917-22 


230-907C6 


65- 13767 


57- 


2551 -75863 


i 179-07078 


50 51493 


•75 


4271-82969 


231-69245 


65-35923 


•25 


2574-19156 


179-85617 


50-73649 


74- 


4300-84034 


232-47785 


65-58079 


•5 


2596-7-2267 


1 180-64157 


50-95304 


•25 


4329-94916 


233-26325 


65-80234 


•75 


2619-35196 


181-4-2697 


51-17960 


•5 


4359-15615 


234-04865 


66-02390 


53- 


264207942 


182 21237 


51-40116 


•75 


433s -46 132 


234-83105 


66-24546 


•25 


2664 90505 


182-99777 


51-62-271 


75^ 


4417 86466 


235-61944 


66-46701 


•5 


2687-82886 


183-78317 


51 844-27 


-25 


4447-36618 


236-40484 


66-68357 


•75 


2710-85084 


184 -56855 


52-06533 


•5 


4476-'..C5-<8 


237-19024 


66-91043 


59^ 


2733-97100 


185-35396 


52-28733 


•75 


4506-66374 


237-97564 


67-13168 


•25 


2757-18933 


186-13936 


52-50394 


76^ 


4536-45979 


233-76104 


67-35324 


•5 


2780-50584 


186-9-2476 


52-73050 


-25 


4566-35100 


239-54643 


67-57480 


•75 


28039-2053 


187-71016 


52 •95-205 


•5 


45.t6 34640 


240-33133 


67-79635 


60- 


23-27-43338 


183-49555 


53-17364 


•75 


46-26-43696 


241-117-23 


63-01791 


•25 


2851-04442 


189-28095 


53 39517 


77- 


4656 6-2571 


241-90-263 


68-23947 


•5 


2874-75362 


19J-06635 


53-61672 


•25 


4636-91-262 


242-63803 


68-46102 


•75 


2898-56100 


190-85175 


53-833-28 


-5 


4717-29771 


243-47343 


68-68253 


61 


2922-46656 


191-63715 


54 05'.''84 


•75 


4747-78093 


244-25332 


68-90414 


•25 


2946-47029 


192-4-2255 


54-28139 


78- 


4773-36-242 


245 04422 


69-12570 


•5 


2970-572-20 


193-20794 


54-50295 


-25 


4809-04204 


245-82962 


69 34725 


•75 


2994-772-23 


193-99334 


54-7-2451, 


•5 


4839-81983 


246-61502 


69-56381 


62- 


3019-07054 


194-77374 


54-94606 


•75 


4370-79579 


247-40042 


69-79037 


•25 


3043-46697 


195^56414 


55-16762 


79^ 


4901-66993 


243-13581 


70-01192 


•5 


3067-96157 


196-34S54 


55-33918 


•25 


4932-74225 


248-97121 


70-23343 


•75 


3092-55435 


197-13493 


5561073 


•5 


4963-91274 


249-75661 


70-45504 


63- 


3117-24531 


197-92033 


55-832-29 


•75 


4995-18140 


250-34201 


70-67659 


•25 


3142-03444 


198-70573 


56-05335, 


80- 


5026-51824 


251-32741 


70-39315 


•5 


3166-92174 


199-49113 


56-27540 


•25 


5053-013-25 


252-11-231 


71-11971 


•75 


3191-90722 


200-27653 


56-49696 


-5 


5039-57644 


252-898-20 


71-341-26 


64- 


3216-9-9U87 


201-06192 


56-71852 


-75 


5121-23781 


253-63350 


71-56282 


•25 


3242-17270 


201-84732 


56-94007 


81- 


5152-99735 


254-46900 


71-78433 


•5 


3-267-45270 


20263272 


57-16163 


-25 


5134-85506 


255-25440 


72-00593 


•~5 


3-292 83088 


203-41812 


57-33319 


•5 


5216-81095 


256-03.^30 


72-2-2749 


65- 


3318-30724 


204-20352 


5760475 , 


•75 


5-243-86501 


256 82579 


72-44905 


•25 


3313-88176 


204-98392 


57-82630 1 


82^ 


5-231 -017-25 


257-61059 


■2-67060 


•5 


3369-55447 


205-77431 


53 04786 


•25 


531326766 


258-39599 


72-89216 


•75 


3395-3-2534 


206-551^71 


53-26942 


•5 


5345-61621 


259-18139 


7311372 


66- 


3421-19439 


207-34511 


53-49097 


•75 


5373-06301 


259-96679 


73-33527 


25 


344716162 


203-13051 


53-71-253 


83^ 


5410-60794 


260-75219 


73-55683 


•5 


3473-227u2 


203-9 1591 


58-93409 


•25 


5443-25105 


261-53753 


73-77839 


•75 


3499 39060 


209-70130 


59-15564 


•5 


5475-99234 


262-32298 


73-99994 


67- 


3525-65235 


210 48670 


59-37720, 


•75 


5508-83180 


263-10338 


74-22150 


25 


3552-01228 


211-27210 


59-59876 


84- 


5541-76944 


263-89378 


74-44306 


•5 


3573-4703S 


212-05750 


59-8-2031! 


•25 


5574-80525 


264-67918 


74-66461 


•75 


360502665 


212-84-290 


60-041871 


•5 


5607-93923 


265-46457 


74-88617 


68- 


3631-68110 


213-62530 


60-26343 


•75 


5641-17139 


266-24997 


75-10773 


•25 


3658-43373 


214-41369 


60-4S498 


85^ 


5674-50173 


267-03537 


75-32923 


•5 


3685-28453 


215-19909 


60-70654 


•25 


5707-93023 


267-82077 


75-55034 


•75 


3712-23350 


215-984491 


60-92810 


•5 


5741-45692 


268-60617 


75-77240 


69- 


3739-28065 


216 76939 


61-14965 


•75 


5775-08178 


269-39157 


75-99395 


•25 


37G6-42597 


217-555-29i 


61-37121 


86^ 


5808-80481: 


270-17696 


76-21551 


•5 


3793-66947 


213340681 


61-59277 


•25 


5842-6-2602' 


270-96236 


76-43707 


•75 


3821-01115 


219- 12608! 


61-81432 


•5 


5876-54540 


271-74776 


76-65362 


70^ 


3348-45100 


219-91148 


62-03538 


•75 


5910-56296 


272-53316 


76-33013 


•25 


3875-98902 


2-20-69683I 


62-25744 


87^ 


5944 67369 


273-31856 


77-10174 


•5 


3903-62522 


22 1 -48-2281 


62-47899 


•25 


5978-89260 


274-10395 


77-32329 


•75 


3931-35959 


2-22-207681 


62-70055 


•5 


6013-20468 


274-88935 


77-54485 


7r 


3959-19214 


223-053071 


62-9-2211 


•75 


6047-61494 


275-67475 


77-76641 


•25 


3987-12286 


223 83347 1 


63-14366 


88^ 


6082-12337 


276-46015 


77-98796 



28- 



APPENDIX. 









Side of 








Side of 


Diam. 


Area. 


Circum. 


equal sq. 


Diam. 


Area. 


Circum. 


equal sq. 


88-25 


6116-72998 


277-24555 


78-20952 


"94"^ 


6976-74097 


296-09510 


83-52688 


•5 


6 151 •43476 


278-03094 


78-43103 


•5 


7013-80194 


296-88050 


83-74<944 


•75 


6186^23772 


278-81634 


78 •65-263 


•75 


7050-96109 


297-66590 


83-97000 


89- 


6221-13885 


279-60174 


78^87419 


95- 


7';)88-2]842 


298-45130 


84-19155 


•25 


6-256-13815 


280-33714 


79-09575 


-25 


7125-57992 


299-23670 


84-41311 


•5 


6291-23563 


281-17254 


79-31730 


•5 


7163-02759 


300-02209 


84^63467 


•75 


6326-43129 


281-95794 


79-53836 


•75 


7200-57944 


300-80749 


84-85622 


90- 


6361-72512 


282-74333 


79-76042 


96- 


7-238-22947 


301-59289 


85-07778 


•25 


6397-11712 


283-52873 


79-98198 


.4/0 


7-275-97767 


302-37829 


85-29934 


•5 


6432-60730 


284-31413 


80-20353 


•5 


7313-82404 


303-16369 


85-52089 


•75 


6468-19566 


285-09953 


80-1-250'J 


•75 


7351-76859 


303-9 1908 


85- 74-245 


91- 


6503-88219 


285-8^493 


80-646(39 


97- 


7389-81131 


304-73448 


85-96401i 


.25 


6539-66689 


286-67032 


80-86820 


•25 


7427-95221 


305-51988 


86-18556 


•5 


6575-51977 


2S7-45572 


81-08976 


-5 


7466-19129 


30(5-30528 


86-40712 


•75 


6611-53082 


288-24112 


81-31132 


•75 


7504-52853 


.307-09068 


86-6-2868 


92^ 


6547-61005 


289-02652 


81-53^87 


93^ 


7542-96396 


307-87(')08 


86-85023 


•25 


6683-78745 


28.)-8il92 


81-75443 


•25 


7581-49755 


308-66147 


87-07179 


•5 


67-20-06303 


290-597:;-2 


8I-975J9 


•5 


7620-12933 


309-44637 


87-29335 


•75 


6756-43678 


291-3^271 


82] 9754 


•75 


765S-85'.i27 


310-23227 


87-51490 


93- 


6792-90871 


292-16811 


82-41910 


99^ 


7(;vi7-68739 


311-01767 


87-73646 


•25 


68-29-478S1 


292-9535] 


8201066 


■25 


7736-61369 


3 11 -80307 


87-95802 


•5 


6866-14709 


293-7389] 


82-86221 


•5 


7775-63816 


312-58846 


88^ 17957 


•75 


6902-91354 


2'.)4-5243i 


83-08377 


•75 


7814-76081 


313-37336 


88-40113 


9^ 


6939-77817 


295 -30970 


83 3i.''5;jo 


!100- 


7-^53-98iG3 


314-159-26 


88-6-2-269 



The followinsf rules are for extfTidiiia the use of the above table. 

To jind the area, circumference^ or side of equal square, of a circle 
having a diameter of more than 100 inches, feet, SfC. Rule. — Divide 
the given diameter by a number that will give a quotient equal to some 
one of the diameters in the table ; then the circumference or side of 
equal square, opposite that diameter, multiplied by tliat divisor, or, the 
area opposite that diameter, multiplied by the square of the aforesaid 
divisor, will give the answer. 

Example. — What is the circumference of a circle whose diameter is 
228 feet ? 228, divided by 3, gives 76, a diameter of the table, the cir- 
cumference of which is 238-761, therefore : 

238-761 
3 



716-283 feet. Ans. 
Another example. — What is the area of a circle having a diameter 
of 150 inches ? 150, divided by 10, gives 15, one of the dianneters in 
the table, the area of which is 176-71458, therefore : 

176-71458 

100=- 10 X 10 



17,671-45800 inches. Ans. 
To fnd the area, circumference, or side of equal square, of a circle 
having an intermediate diameter to those in the tahle. Rule. — Multiply 
the given diameter by a number that will give a product equal to some 
one of the diameters in the table ; then the circumference or side of 
equal square opposite that diameter, divided by that multiplier, or, the 
area opposite that diameter divided by the .square of the aforesaid mul- 
tiplier, will give the answer. 



APPENDIX. 



29 



Example. — What is the circumference of a circle whose diameter is 
6^, or 6-125 inches ? 6-125, multiplied by 2, gives 12-25, one of the 
diameters of the table, whose circumference is 3S-484, therefore : 

2)38-484 

19-242 inches. Ans. 
Another example. — What is the area of a circle, the diameter of 
which is 3-2 feet"? 3-2, multiplied by 5, gives 16, and the area of 16 
is 201-0619, therefore : 

5 X 5 — 25)201-0619(8-0424 + feet. Ans. 
200 

106 
100 



61 
50 

119 
100 

19 
Note. — The diameter of a circle, multiplied by 3-14159, vv^ill give 
its circumference ; the square of the diameter, multiplied by -78539, 
will give its area ; and the diameter, multiplied by -88622, will give 
the side of a square equal to the area of the circle. 



TABLE SHOWING THE CAPACITY OF WELLS, CISTERNS, &C. 



The gallon of the state of New- York is required to contain 8 pounds of pure water ; and 
since a cubic foot of pure water weighs 62-5 pounds, the gallon contains 2211S4 cubic 
inches. Upon these data the following table is computed. 

One foot in depth of a cistern of 
3 feet diameter will contain 



H 


do. 


do. 


4 


do. 


do. 


H 


do. 


do. 


5 


do. 


do. 


5i 


do. 


do. 


6 


do. 


do. 


6i 


do. 


do. 


7 


do. 


do. 


8 


do. 


do. 


9 


do. 


do. 


10 


do. 


do. 


12 


do. 


do. 





- 55-223 


gallons 


- 


75-164 


do. 




- 98-174 


do. 


- 


124-252 


do. 




. 153-39 


do. 


. 


185-611 


do. 




- 220-893 


do. 


. 


259-242 


do. 




- 300-66 


do. 


. 


392-699 


do. 




. 497-009 


do. 


. 


613-592 


do. 


- 


- 883-573 


do. 



Note. — To reduce cubic feet to gallons, divide by the decimal, 
128. 



TABLE OF POLYGONS. 

(From Gregory's Mathematics.) 





Names. 


Multipliers for 
areas. 


Radius of cir- 
cum. circle. 


Factors for 
sides. 


3 


Trigon 


0-4330127 


0-5773503 


1-732051 


4 


Tetragon, or Square 


l-OOOOOOO 


0-7071068 


1-414214 


5 


Pentagon - 


1-7204774 


0-8506508 


1-175570 


6 


Hexagon 


2-5980762 


1-0000000 


1-000000 


7 


Heptagon - 


3-6339124 


1-1523824 


0-867767 


8 


Octagon 


4-8284271 


1-3065628 


0-765367 


9 


Nonagon - 


6-1818242 


1-4619022 


0-684040 


10 


Decagon 


7-6942088 


1-6180340 


0-618034 


11 


Undecagon 


9-3656399 


1-7747324 


0-503465 


12 


Dodecagon 


11-1961524 


1-9318517 


0-517638 



To Jind the area of any regular polygon, ivhose sides do not exceed 
twelve. Rule. — Multiply the square of" a side of the given polygon by 
the number in the column termed Multipliers for areas, standing op- 
posite the name of the given polygon, and the product will be the an- 
swer. Example. — What is the area of a regular heptagon, whose 
sides measure each 2 feet ? 

3-6339124 

4 = 2X2 



14-5356496: Ans. 
To jind the radius of a circle which will circumscrihe any regular 
polygon given, whose sides do not exceed twelve. Rule. — Multiply a 
side of the given polygon by the number in the column termed Radius 
of circumscrihing circle, standing opposite the name of the given poly- 
gon, and the product will give the answer. Example. — What is the 
radius of a circle which will circumscribe a regular pentagon, whose 
sides measure each 10 feet ? 

•8506508 
10 



8-5065080: Ans. 
To find the side of any regular polygon that may he inscribed within 
a given circle. Rule. — Multiply the radius of the given circle by the 
number in the column termed Factors for sides, standing opposite the 
name of the given polygon, and the product will be the answer. Ex- 
ample. — What is the side of a regular octagon that may be inscribed 
within a circle, whose radius is 5 feet ? 

•765367 



' 3-826835: Ans. 



WEIGHT OF MATERIAI^. 



U)s. in a 
Woods. cubic foot. 

Apple, . - - - 49 

Ash, . - - 45 

Beach, - - - - 40 

Birch, ... 45 

Box, .... 60 

Cedar, . - - 28 

Virginian red cedar, - 40 

Cherry, ... 38 

Sweet chestnut, - - 36 

Horse-chestnut, - - 34 

Cork, .... 15 

Cypress, - - - 28 
Ebony, - - - -83 

Elder, - - - 43 

Elm, - . - - 34 

Fir, (white spruce,) - 29 

Hickory, - - - 52 

Lance-wood, - - 59 

Larch, - - - - 31 

Larch, (whitewood,) - 22 

Lignum-vitae, - - - 83 

Logwood, ... 57 

St. Domingo mahogany, - 45 

Honduras, or bay mahogany, 35 

Maple, - - - . 47 

White oak, - - 43 to 53 

Canadian oak, - - 54 

Red oak, . - - 47 

Live oak, - - - 76 

White pine, - - 23 to 30 

Yellow pine, - 34 to 44 

Pitch pine, - - 46 to 58 

Poplar, - - - 25 

Sycamore, - - - 36 

Walnut, - - 40 



Metals. cubic foot. 

Wire-drawn brass, - 534 

Cast brass, - , - 506 

Sheet-copper, - - 549 

Pure cast gold, - - 1210 
Bar-iron, - 475 to 487 

Cast iron, - - 450 to 475 
Milled lead, - - .713 
Cast lead, - - 709 

Pewter, - - - 453 

Pure platina, - - 1345 
Pure cast silver, - - 654 
Steel, - - 486 to 490 

Tin, - - - - 456 
Zinc, - - - 439 

Stone, Earths, cf*c. 

Brick, Phila. stretchers, 105 
North river common hard 
brick, - - - 107 

Do. salmon brick, 100 

Brickwork, about - 95 

Cast Roman cement, - 100 
Do. and sand in equal parts, 113 
Chalk, - 144 to 166 

Clay, - - - - 119 
Potter's clay, - 112 to 130 
Common earth, 95 to 124 

Flint, . - - - 163 
Plate-glass, - - 172 

Crown-glass, ... 157 
Granite, - - 158 to 187 
Quincy granite, - - 166 
Gravel, - - - 109 
Grindstone, - - - 134 
Gvpsum, (Plaster-stone,) 142 
Unslaked lime, - - 52 



32 



! APPENDIX. 




ths. 


in a 




lbs. in a 


cubic foot. 




cvinc foot. 


Limestone, - - 11-8 to 198 


Common blue stone. 


160 


Marble, - - 161 to 177 


Silver-gray flagging. 


- 185 


New mortar, - 


107 


Stonework, about. 


120 


Dry mortar, 


90 


Common plain tiles. 


- 115 


Mortar with hair, (Plaster- 




Sundries. 




ing,) ... - 


105 


Atmospheric air. 


- 0-075 


Do. dry, 


86 


Yellow beesxvax, - 


- 60 


Do. do. including Jath 




Birch-charcoal, - 


34 


and nails, from 7 to 11 




Oak-charcoal, 


- 21 


lbs. per superficial foot. 




Pine-charcoal, 


17 


Crystallized quartz. 


165 


Solid gunpowder, - 


- 109 


Pure quartz-sand. 


171 


Shaken gunpowder. 


58 


Clean and coarse sand. 


100 


Honey, 


. 90 


Welsh slate, - - - 


180 


Milk, 


64 


Paving stone, 


151 


Pitch, - 


. 71 


Pumice stone. 


56 


Sea-water, 


64 


Nyack brown stone, - 


148 


Rain-water, - 


- 62-5 


Connecticut brown stone, 


170 


Snow, 


8 


Tarrytown blue stone, - 


171 


Wood.ashes, 


- 58 



ROOTS OF DECIMALS. 

Rule. — Seek for the given decimal in the column of numbers, and opposite in the 
columns of roots will be found the answer, correct as to the figures, but requiring 
the decimal point to be shifted. The transposition of the decimal point is to be per- 
formed thus : For every place the decimal point is removed in the root, remove it in 
the number two places for the sq^iare root and three places for the cube root. 

Examples. — By the table, the square root of 86-0 is 9'2736, consequently, by the 
rule the square root of 0*86 is 0-92736. The square root of 9- is 3-, hence the square 
root of 0"09 is 0-3. For the square root of 0*0657 we have 0-25fi? " found opposite 
No. 657. So, also, the square root of 0-000927 is 0-030446, foui. -^site No. 927. 

And the square root of 873 (whole number with decimals) is 2-95-- 1 opposite 

No. 873. The cube root of 0-8 is 0-928, found at No. 800 ; the cube -f 0-08 is 

0-4308, found opposite No. 80, and the cube root of 0-008 is 0*2, as 2-0 is the Cal)e 
root of 8-0. So also the cube root of 0-047 is 0-36088, found opposite No. 47. 



THE END. 



\ 



iN 589 



Q-) 



